biologically inspired adaptive dynamic walking of a
TRANSCRIPT
paper.dviof a Quadruped Robot
Hiroshi Kimura∗ Yasuhiro Fukuoka∗ Avis H. Cohen∗∗ ∗Graduate School of Information Systems
Univ. of Electro-Communications 1-5-1 Chofugaoka, Chofu Tokyo 182-8585, Japan
{hiroshi, fukuoka}@kimura.is.uec.ac.jp
Univ. of Maryland, College Park MD 20742, USA [email protected]
Abstract
We have been trying to induce a quadruped robot to walk with medium walking speed on ir- regular terrain based on biological concepts. We propose the necessary conditions for stable dy- namic walking on irregular terrain in general, and we design the mechanical system and the neu- ral system by comparing biological concepts with those necessary conditions described in physical terms. PD-controller at joints constructs the vir- tual spring-damper system as the visco-elasticity model of a muscle. The neural system model con- sists of a CPG (central pattern generator), re- flexes and responses. A CPG generates rhythmic motion for walking. A CPG receives sensory in- put and changes the period of its own active phase as responses. The virtual spring-damper system also receives sensory input and outputs torque as reflexes. The states in the virtual spring-damper system are switched based on the phase signal of the CPG. The physical oscillations such as the motion of the virtual spring-damper system of each leg and the rolling motion of the body are mutually entrained with the neural oscillations of CPGs. Consequently, the adaptive walking is generated by the interaction with environment. In this paper, we report our experimental results of dynamic walking on terrains of medium degrees of irregularity in order to verify the effectiveness of the designed neuro-mechanical system.
1. Introduction
Many previous studies of legged robots have been per- formed, including studies on running (Hodgins and Raib- ert 1991) and dynamic walking (Yamaguchi, Takanishi and Kato 1994; Kajita and Tani 1996; Chew, Pratt and Pratt 1999; Yoneda, Iiyama and Hirose 1994; Buehler et al. 1998) on irregular terrain. However, all of those studies assumed that the structure of terrain was known,
even though the height of the step or the inclination of the slope was unknown.
On the other hand, studies of autonomous dynamic adaptation allowing a robot to walk over irregular ter- rain with less knowledge of it have been started only re- cently and by only a few research groups. One example is the recent achievement of high-speed mobility of a hexa- pod over irregular terrain, with appropriate mechanical compliance of the legs (Saranli, Buehler and Koditschek 2001; Cham et al. 2002). The purpose of this study is to realize high-speed mobility on irregular terrain us- ing a mammal-like quadruped robot, the dynamic walk- ing of which is less stable than that of hexapod robots, by referring to the marvelous abilities of animals to au- tonomously adapt to their environment.
As many biological studies of motion control pro- gressed, it has become generally accepted that animals’ walking is mainly generated at the spinal cord by a combination of a CPG (central pattern generator) and reflexes receiving adjustment signals from a cerebrum, cerebellum and brain stem (Grillner 1981; Cohen and Boothe 1999). A great deal of the previous research on this attempted to generate walking using a neural system model, including studies on dynamic walking in simulation (Taga, Yamaguchi and Shimizu 1991; Taga 1995; Miyakoshi et al. 1998; Ijspeert 1998, 2001), and real robots (Ilg et al. 1999; Tsujita, Tsuchiya et al. 2001; Lewis et al. 2003). But autonomously adaptive dynamic walking on irregular terrain was rarely realized in those earlier studies except for our studies (Kimura et al. 1999, 2001; Fukuoka et al. 2003). This paper reports on our progress in the past couple of years using a newly de- veloped quadruped called “Tekken1&2,” which contains a mechanism designed for walking in 3D space (pitch, roll and yaw planes) on irregular terrain. Especially, the self-contained (power autonomous) system was realized in Tekken2 (Fig.1) and we succeeded in outdoor experi- ments using it.
Table 1: Biological concepts of legged locomotion control.
Limit-Cycle-based Control ZMP-based Control by Neural System by Mechanism
(CPG and reflexes) (spring and damper) good for posture and low medium speed walking high speed running control of speed walking
main upper neural system lower neural system musculoskeletal system controller acquired by learning (at spinal cord, brain stem, etc.) through self stabilization
battery
controller
battery
radio receiver
Figure 1: Self-contained quadruped robot: Tekken2. The
length of the body and a leg in standing are 30 [cm] and
20 [cm]. The weight including batteries is 4.3 [kg]. The hip
pitch joint, knee pitch joint and hip yaw joint are activated
by DC motors of 23 [W], 23 [W] and 8 [W] through gear ratio
of 20, 28 and 18, respectively.
2. Design concepts for adaptive walking
2.1 Legged locomotion control
Methods for legged locomotion control are classified into ZMP-based control and limit-cycle-based control (Table 1). ZMP (zero moment point) is the extension of the cen- ter of gravity considering inertia force and so on. It was shown that ZMP-based control is effective for control- ling posture and low-speed walking of a biped (Takan- ishi et al. 1990) and a quadruped (Yoneda, Iiyama and Hirose 1994). However, ZMP-based control is not good for medium or high-speed walking from the standpoint of energy consumption, since a body with a large mass needs to be accelerated and decelerated by actuators in every step cycle.
In contrast, motion generated by the limit-cycle-based control has superior energy efficiency. But there exists the upper bound of the period of the walking cycle, in which stable dynamic walking can be realized (Kimura et al. 1990). It should be noted that control by a neural system consisting of CPGs and reflexes is dominant for various kinds of adjustments in medium-speed walking of animals (Grillner 1981). Full and Koditschek (1999)
also pointed out that, in high-speed running, kinetic en- ergy is dominant, and self-stabilization by a mechanism with a spring and a damper is more important than ad- justments by the neural system. Our study is aimed at medium-speed walking controlled by neural system con- sisting of CPGs and reflexes (Table 1).
2.2 Necessary conditions for stable dynamic walking on irregular terrain
We propose the necessary conditions for stable dynamic walking on irregular terrain, which can be itemized in physical terms:
(a) the period of the walking cycle should be shorter enough than the upper bound of it, in which stable dynamic walking can be realized,
(b) the swinging legs should be free to move forward during the first period of the swing phase,
(c) the swinging legs should land reliably on the ground during the second period of the swing phase,
(d) the angular velocity of the supporting legs relative to the ground should be kept constant during their pitching motion or rolling motion around the contact points at the moment of landing or leaving,
(e) the phase difference between rolling motion of the body and pitching motion of legs should be main- tained regardless of a disturbance from irregular ter- rain, and
(f) the phase differences between the legs should be maintained regardless of delay in the pitching mo- tion of a leg receiving a disturbance from irregular terrain.
We design the neural system for these necessary condi- tions to be satisfied in order to realize adaptive walking.
2.3 Mutual entrainment between neural system and mechanical system
Motion generation based on biological concepts is illus- trated as Fig.2, where a neural system and a mechanical system have their own non-linear dynamics. The char- acteristic of this method is that there is no adaptation through motion planning. These two dynamic systems
are coupled to each other, generating motion by inter- acting with the environment emergingly and adaptively (Taga, Yamaguchi and Shimizu 1991; Taga 1995). We call this method “coupled-dynamics-based motion gen- eration.”
The coupled dynamic system can induce autonomous adaptation according to its own dynamics, under changes in the environment (e.g., adaptive walking on ir- regular terrain) and under adjustment of the neural sys- tem parameters by an upper-level controller (e.g., gait transition in change of walking speed). Therefore, we can avoid such serious problems in robotics as model- ing of mechanical system and environment, autonomous planning, conflict between planned motion and actual motion and so on.
neural system
2.4 Design of neural system
We construct the neural system centering a neural oscil- lator as a model of a CPG, since the exchange between the swing and stance phases in the short term and the quick adjustment of these phases on irregular terrain are essential in the dynamic walking of a quadruped where the unstable two-legged stance phase appears. On the other hand, Espenschied et al. (1996) constructed the gait pattern generator proposed by Cruse (1990) refer- ring to a stick insect. They also employed the sway- ing, stepping, elevator and searching reflexes observed by Pearson et al. (1984) in a stick insect. Consequently, their neural system is more sensor dependent and more decentralized by using a non-oscillator type CPG.
The characteristics of our neural system in comparison with the one used by Espenschied et al. (1996) are:
(1) The cyclic period is mainly determined by the time constant of CPGs. This makes it easy for the nec- essary condition (a) described in Section 2.2 to be satisfied.
(2) The gait is mainly determined by the connecting weights of the CPGs network.
(3) As sensor feedback for adaptation on irregular ter- rain, a “response” directly and quickly modulating
the CPG phase is employed in parallel with a “re- flex” directly generating joint torque (Fig.2).
About the characteristic (3), it is well known in physiol- ogy that
• some sensory stimuli modify CPG activity and reflex- ive responses to sensory stimuli are phase dependent under CPG activity (Cohen and Boothe 1999).
Such interaction between CPG activity and a sensory stimulus is very important for adaptation, and corre- sponds to the necessary conditions described in physical terms in Section 2.2.
2.5 Design of mechanical system
Each leg of Tekken has a hip pitch joint, a hip yaw joint, a knee pitch joint, and an ankle pitch joint (Fig.1). The direction in which Tekken walks can be changed by using the hip yaw joints. Two rate gyro sensors and two incli- nometers are mounted on the body in order to measure the body pitch and roll angles.
In order to obtain appropriate mutual entrainment be- tween neural system and mechanical system, mechanical system should be well designed to have the good dy- namic properties. In addition, performance of dynamic walking such as adaptability on irregular terrain, energy efficiency, maximum speed and so on highly depends on the mechanical design. The design concepts of Tekken are:
[a] high power actuators and small inertia moment of legs for quick motion and response,
[b] small gear reduction ratio for high backdrivability to increase passive compliance of joints,
[c] small mass of the lowest link of legs to decrease im- pact force at collision,
[d] small contacting area at toes to increase adaptability on irregular terrain,
[e] passive ankle joint mechanism to prevent a swinging leg from stumble on an obstacle quickly (Fig.3).
A
Figure 3: The ankle joint can be passively rotated:(A) if
the toe contacts with an obstacle in a swing phase, and is
locked:(B) while the leg is in a stance phase.
3. Implementation of neural system for adaptive walking
3.1 Rhythmic motion by CPG
Although actual neurons as a CPG in higher animals have not yet become well known, features of a CPG have been actively studied in biology, physiology, and so on. Several mathematical models were also proposed, and it was pointed out that a CPG has the capability to gener- ate and modulate walking patterns and to be mutually entrained with a rhythmic joint motion (Grillner 1981; Cohen and Boothe 1999; Taga, Yamaguchi and Shimizu 1991; Taga 1995). As a model of a CPG, we used a neu- ral oscillator proposed by Matsuoka (1987), and applied to the biped simulation by Taga et al. (1991; 1995). A single neural oscillator consists of two mutually inhibit- ing neurons (Fig.4-(a)). Each neuron in this model is represented by the following nonlinear differential equa- tions:
τu{e,f}i = −u{e,f}i + wfey{f,e}i − βv{e,f}i
+u0 + Feed{e,f}i + n∑
j=1
wijy{e,f}j
y{e,f}i = max (u{e,f}i, 0) (1) τ ′v{e,f}i = −v{e,f}i + y{e,f}i
where the suffix e, f , and i mean an extensor neuron, a flexor neuron, and the i-th neural oscillator, respectively. u{e,f}i is uei or ufi, that is, the inner state of an extensor neuron or a flexor neuron of the i-th neural oscillator; v{e,f}i is a variable representing the degree of the self- inhibition effect of the neuron; yei and yfi are the output of extensor and flexor neurons; u0 is an external input with a constant rate; Feed{e,f}i is a feedback signal from the robot, that is, a joint angle, angular velocity and so on; and β is a constant representing the degree of the self- inhibition influence on the inner state. The quantities τ and τ ′ are time constants of u{e,f}i and v{e,f}i; wfe is a connecting weight between flexor and extensor neurons; wij is a connecting weight between neurons of the i-th and j-th neural oscillator.
In Fig.4-(a), the output of a CPG is a phase signal: yi
. yi = −yei + yfi (2)
The positive or negative value of yi corresponds to ac- tivity of a flexor or extensor neuron, respectively.
We use the following hip joint angle feedback as a basic sensory input to a CPG called a “tonic stretch response” in all experiments of this study. This negative feedback makes a CPG be entrained with a rhythmic hip joint motion.
Feede·tsr = ktsr(θ − θ0), F eedf ·tsr = −Feede·tsr (3)
Feed{e,f} = Feed{e,f}·tsr (4)
Neural Oscillator(a)
LF : left fore leg
LH : left hind leg RF : right fore leg RH : right hind leg
(b) Neural Oscillator Network
extensor neuron of other N.O.’s
flexor neuron of other N.O.’s
Figure 4: Neural oscillator as a model of a CPG. The suffix
i, j = 1, 2, 3, 4 corresponds to LF, LH, RF, RH. L, R, F or H
means the left, right, fore or hind leg, respectively.
where θ is the measured hip joint angle, θ0 is the origin of the hip joint angle in standing and ktsr is the feedback gain. We eliminate the suffix i when we consider a single neural oscillator.
By connecting the CPG of each leg (Fig.4-(b)), CPGs are mutually entrained and oscillate in the same period and with a fixed phase difference. This mutual entrain- ment between the CPGs of the legs results in a gait. The gait is a walking pattern, and can be defined by phase differences between the legs during their pitching motion. The typical symmetric gaits are a trot and a pace. Diagonal legs and lateral legs are paired and move together in a trot gait and a pace gait, respectively. A walk gait is the transversal gait between the trot and pace gaits. We used a trot gait for most of experiments. But the autonomous gait transition in changing walking speed was discussed in our former study (Fukuoka et al. 2003).
3.2 Virtual spring-damper system
We employ the model of the muscle stiffness, which is generated by the stretch reflex and variable according to the stance/swing phases, adjusted by the neural system. The muscle stiffness is high in a stance phase for sup- porting a body against the gravity and low in a swing phase for compliance against the disturbance (Akazawa et al. 1982). All joints of Tekken are PD controlled to move to their desired angles in each of three states (A, B, C) in Fig.5 in order to generate each motion such as swinging up (A), swinging forward (B) and pulling down/back of a supporting leg (C). The timing for all joints of a leg to switch to the next state are:
A→ B: when the hip joint angle of the leg reaches the desired angle of the state (A)
B → C: when the CPG extensor neuron of the leg becomes active (yi ≤ 0)
C → A: when the CPG flexor neuron of the leg becomes active (yi > 0)
A
Figure 5: State transition in the virtual spring-damper sys-
tem. The desired joint angles in each state are shown by the
broken lines.
The desired angles and P-gain of each joint in each state are shown in Table 2, where constant values of the desired joint angles and constant P-gains were de- termined through experiments. Since Tekken has high backdrivability with small gear ratio in each joint, PD- controller can construct the virtual spring-damper sys- tem with relatively low stiffness coupled with the me- chanical system. Such compliant joints of legs can im- prove the passive adaptability on irregular terrain.
3.3 CPGs and pitching motion of legs
The diagram of the pitching motion control consisting of CPGs and the virtual spring-damper system is shown in the middle part of Fig.6. Joint torque of all joints is determined by the PD controller, corresponding to a stretch reflex at an α motor neuron in animals. The desired angle and P-gain of each joint is switched based on the phase of the CPG output: yi in Eq.(2) as described in Section 3.2. As a result of the switching of the virtual spring-damper system and the joint angle feedback signal to the CPG in Eq.(4), the CPG and the pitching motion of the leg are mutually entrained.
3.4 Reflexes and responses
It is known in physiology that the reflex to a stimu- lus on the paw dorsum in the walking of a cat depends on whether flexor or extensor muscles are active (as re- viewed in Cohen and Boothe 1999). That is,
• when flexor muscles are active, the leg is flexed in order to escape from the stimulus,
Table 2: Desired value of the joint angles and P-gains at the
joints used in the PD-controller for the virtual spring-damper
system in each state shown in Fig.5. θ, φ and ψ are the hip
pitch joint angle, the knee pitch joint angle and the hip yaw
joint angle, respectively.
P control angle in state desired value[rad] P-gain[Nm/rad]
θ in A 1.2θC→A G1
θ in B −0.17 G2v+G3
θ in C θstance+ −G4v+G5
body pitch angle
ψ in all states 0 G8
θC→A: the hip joint angle measured at the instance when the state changes from (C) to (A).
θstance: variable to change the walking speed. body pitch angle: the measured pitching angle of
the body used for the vestibulospinal reflex. ∗ means that the desired angle is calculated
on-line for the height from the toe to the hip joint to be constant.
v [m/s]: the measured walking speed of Tekken.
• when extensor muscles are active, the leg is strongly extended in order to prevent the cat from falling down.
We call these a “flexor reflex” and a “extensor reflex,” respectively, and we assume that the phase signal from the CPG of the leg switches such reflexes.
The following biological concepts are also known (Ogawa et al. 1998):
• when the vestibule in a head detects an inclination in pitch or roll plane, a downward-inclined leg is ex- tended while an upward-inclined leg is flexed (Fig.7).
We call this a “vestibulospinal reflex.”
Referring to such biological knowledge, we employed the several reflexes and responses (Fig.6) to satisfy the necessary conditions (b)∼(e) described in physical terms in Section 2.2 in addition to the stretch reflex and re- sponse described in Section 3.2 and 3.1. The necessary condition (f) can be satisfied by the mutual entrainment between CPGs and the pitching motion of legs, and the mutual entrainment among CPGs (Kimura et al. 2001).
3.4.1 Flexor and extensor reflexes
The flexor and extensor reflexes contribute to satisfy the conditions (b) and (c), respectively. In our former stud- ies (Kimura et al. 1999, 2001), the stumble of a swinging
- θ swing
α
Figure 6: Control diagram for Tekken. PD-control at the hip
yaw and knee pitch joints are eliminated in this figure.
leg on an obstacle was detected by force sensor, and the flexor or extensor reflex was activated afterwards. How- ever, the problem was that the robot typically fell down due to the delayed flexing motion caused by the delay of sensing and large inertia of the leg while walking with short cyclic period. Therefore, we substitute the flexor reflex for the passive ankle joint mechanism (Fig.3) uti- lizing the fact that the collision with a forward obstacle occurs in the first half of a swing phase. The extensor reflex has not yet been implemented in Tekken.
3.4.2 Vestibulospinal reflex and response for pitching
We call the reflex/response for an inclination in the pitch plane a “vestibulospinal reflex/response for pitching,” the role of which corresponds to the condition (d) and (f). In Tekken, hip joint torque in the stance phase is adjusted by the vestibulospinal reflex (Fig.7-(a)), since the body pitch angle is added to θstance in Table 2. For the vestibulospinal response, the following equations are used rather than Eq.(3), (4).
θvsr = θ − (body pitch angle) Feed{e,f}·tsr·vsr = ±ktsr(θvsr − θ0).
(5)
roll plane
upward -inclined
3.4.3 Vestibulospinal reflex and response for rolling
We call the reflex/response for an inclination in the roll plane a “tonic labyrinthine reflex/response1 .”
• Tonic labyrinthine reflex
The tonic labyrinthine reflex (TLRF) is employed as the adjustment of P-gain of the knee joint of the sup- porting legs (G7 in Table 2) according to Eq.(7).
G7 = δ(leg) ktlrf × (body roll angle) +G7 (7)
δ(leg) = {
1, if leg is a right leg; −1, otherwise
When an inclination of a body in roll plane (body roll an- gle) is detected, the knee joint P-gain of the downward- inclined legs is increased to extend those legs. In addi- tion, the knee joint P-gain of the upward-inclined legs is decreased to flex those lges. As a result, the inclination of the body in roll plane is decreased (Fig.7-(b)).
• Tonic labyrinthine response
On the other hand, the tonic labyrinthine response (TLRS) is employed as rolling motion feedback to CPGs (upper left part of Fig.6). In Tekken, we made the body roll angle be inputted to the CPGs as a feedback signal expressed by Eq.(8), (9).
Feede·tlrs = δ(leg) ktlrs × (body roll angle) Feedf ·tlrs = −Feede·tlrs
(8)
Feedf = Feedf ·tsr·vsr + Feedf ·tlrs (9)
1The “tonic labyrinthine reflex” is defined in Ogawa et al. (1998). The same reflex is called “vestibular reflex” in Ghez (1991).
The rolling motion feedback to CPGs: Eq.(8), (9) con- tributes to an appropriate adjustment of the periods of the stance and swing phases while walking on irregular terrain (Fukuoka et al. 2003).
The rolling motion feedback to CPGs is important also in order to satisfy the condition (e). CPGs, the pitch- ing motions of the legs, and the rolling motion of the body are mutually entrained through the rolling mo- tion feedback to CPGs (Fig.8). This means that the rolling motion can be the standard oscillation for whole oscillations, in order to compensate for the weak connec- tion between the fore and hind legs in the CPG network (Fig.4-(b)). As a result, the phase difference between the fore and hind legs is fixed, and the gait becomes stable.
Excitatory Connection Inhibitory Connection
angle
Figure 8: Relation among CPGs, pitching motion of a leg
and rolling motion of a body
3.4.4 Stepping reflex to stabilize rolling motion
It is known that the adjustment of the sideway touch- down angle of a swinging leg is effective in stabiliz- ing rolling motion against disturbances (Miura and Shi- moyama 1994). We call this a “sideway stepping reflex,” which helps to satisfy the condition (d) during rolling motion. The sideway stepping reflex is effective also in walking on a sideway inclined slope.
For examples, when Tekken walks on a right-inclined slope (Fig.9), Tekken continues to walk while keeping the phase differences between left and right lges with the help of the tonic labyrinthine response. But Tekken can- not walk straight and shifts its walking direction to the right due to the difference of the gravity load between left and right legs. In addition, Tekken typically falls down to the right for the perturbation from the left in the case of Fig.9-(a), since the wide stability margin: WSM 2 is small. The sideway stepping reflex helps to stabilize the
2the shortest distance from the projected point of the center of gravity to the edges of the polygon constructed by the projected points of legs independent of their stance or swing phases (Fukuoka et al. 2003)
walking direction and to prevent the robot from falling down while keepingWSM large on such sideway inclined slope (Fig.9-(b)).
front view
WSM WSM
(a) (b)
a sideway stepping reflex, (b):With.
Since Tekken has no joint round the roll axis, the side- way stepping reflex is implemented as changing the de- sired angle of the hip yaw joint from 0 (Table 2) to ψ∗
according to Eq.(10).
ψ∗ = δ(leg) kstpr × (body roll angle) (10)
3.4.5 Stepping reflex to stabilize forward speed
In Table 2, P-gain of hip joints in the state (B) and (C) is adjusted using the measured walking speed: v. For examples, when v is increased in the state (B) as the swing phase, the swinging legs land on the forwarder position because P-gain is increased and the swinging legs are more strongly pulled forward. As a result, the increase of v is depressed since the forward motion of the inverted pendulum in the next stance phase is depressed by the gravity (Miura and Shimoyama 1994). On the other hand, when v is decreased in the state (C) as the stance phase while walking up a slope, the increase of P-gain generates additional torque against the gravity load at hip joints of the supporting legs. As a result, the decrease of v is depressed. These reflexes contribute to satisfy the condition (d).
3.4.6 Re-stepping reflex and response for walk- ing down a step
When loss of ground contact is detected in a swing phase while walking over a ditch, a cat activates re-stepping to extend the swing phase and make the leg land on the forwarder position (Hiebert et al. 1994). We call this “re-stepping reflex/response,” which is effective for the necessary condition (c) and (d) to be satisfied also in walking down a large step (Fig.10).
re-stepping
4. Experiments
4.1 Experiments under instantaneous distur- bances
We made Tekken1 walk on several terrains of medium de- gree of irregularity in indoor environment (Fig.11) with all responses and five reflexes, those are, the stretch reflex, flexor reflex, vestibulospinal reflex for pitching, stepping reflex for forward speed and re-stepping re- flex, among seven reflexes described in Section 3.4. This means that Tekken can cope with the instantaneous dis- turbances while walking over irregular terrain consist- ing of series of steps and short slopes. Without a tonic labyrinthine response, the gait was greatly disturbed on irregular terrain, even if Tekken1 didn’t fall down.
(a) (b)
(c) (d)
Figure 11: Walking over a step 4 [cm] in height: (a), walking
up and down a slope of 10 [deg] in a forward direction: (b),
walking over pebbles: (c) and walking over sideway slopes of
3 and 5 [deg]: (d)
Especially noteworthy, Tekken1 successfully walked down a large step with approx. 0.5 [m/s] speed using the re-stepping reflex/response (Fig.12). Without the re-stepping reflex/response, Tekken1 typically fell down
6.0 6.5 7.0 7.55.5
flexor
extensor
0.14[s]
re-stepping response
Figure 12: Walking down a step of 7 [cm] in height. A re-
stepping response was activated when the contact of the right
fore leg had not been detected for 0.14 [s] after the activity
of the flexor neuron became zero.
forward because fore legs landed on the backwarder po- sition excessively and could not depress the increased forward speed.
4.2 Experiments under long-lasting distur- bances
We made Tekken2 walk on a right-inclined slope of 4 [deg] (0.07 [rad]) in indoor environment with all re- sponses and reflexes described in Section 3.4 in order to confirm the effectiveness of a tonic labyrinthine re- flex and a sideway stepping reflex (Fig.9-(b)) under long- lasting disturbances. As a result of the experiment, the body roll angle and the hip yaw angle ψ of the right hindleg and left hindleg are shown in Fig.13. Tekken2 had walked on the right-inclined slope for 2 [sec], and walked on flat terrain afterwards.
In Fig.13, we can see that the body roll angle was posi- tive (0.05∼0.14 [rad]) while walking on the right-inclined slope. The hip yaw joint of the right hindleg moved to the outside of the body (right) by approx. 0.04 [rad] due to the sideway stepping reflex in the swing phase and moved to the inside of the body by approx. -0.04 [rad] due to the gravity load in the stance phase. On the other hand, the hip yaw joint of the left hindleg moved to the inside of the body (right) by approx. -0.04 [rad] due to the sideway stepping reflex in the swing phase and moved to the inside of the body furthermore by approx. -0.1 [rad] due to the gravity load in the stance phase. Consequently, Tekken2 succeeded in straight walking on the sideway inclined slope.
4.3 Outdoor experiments using Tekken2
Even on a paved road in outdoor environment, there ex- ist a slope of 3 [deg] at most, bumps of 1 [cm] in height and small pebbles everywhere. With all responses and reflexes described in Section 3.4, Tekken2 successfully maintained a stable gait on the paved road for 60 [sec]
-0.1
-0.05
0.05
0.1
0.15
[rad]
[sec]
body roll angle ψ of right hindleg ψ of left hindleg
inclination of a slope (0.07 [rad])
swing phase
stance phase
swing phase
stance phase
Figure 13: Walking on a right-inclined slope of 4 [deg]
Figure 14: Photos of walking in outdoor environment
with approx. 0.5 [m/s] speed while changing its walk- ing speed and direction by receiving the operation com- mands from the radio controller (Fig.14).
5. Discussion : how to determine the val- ues of parameters
Values of all parameters in the neural system including the virtual spring-damper system except for θstance were determined experimentally. But it should be noted that those values in each robot were fixed in all experiments independent of kinds of terrain. Although the size and weight of Tekken2 are different from those of Tekken1, the values of the parameters of CPGs used for Tekken2 were same with those used for Tekken1.
The most important subject in coupled-dynamics- based motion generation is to design and construct a neural system carefully, while taking into account the dy- namics of a mechanical system and its interaction with the environment. In this study, we designed a neural system consisting of CPGs, responses and reflexes, while taking into account the characteristics of dynamic walk- ing and utilizing knowledge and concepts in physics, bi- ology, physiology and so on described in Section 2. and 3.. The relationship between parameters of CPGs and
the mechanical system was previously analyzed to some extent (Kimura et al. 2001). Since the relationship be- tween parameters of reflexes/responses and the mechan- ical system has not yet been investigated, we manually determined values of those parameters through experi- ments. The issue of how to construct a neural system suitable for a mechanical system corresponds to the is- sue of “embodiment” at the lowest level of sensorimotor coordination.
In addition, we pointed out the trade-off problem be- tween the stability and the energy consumption in deter- mining the cyclic period of walking on irregular terrain, and showed one example to solve this problem (Fukuoka et al. 2003).
6. Conclusion
In the neural system model proposed in this study, the relationships among CPGs, sensory input, reflexes and the mechanical system are simply defined, and motion generation and adaptation are emergingly induced by the coupled dynamics of a neural system and a mechanical system by interacting with the environment. To gener- ate appropriate adaptation, it is necessary to design both the neural system and the mechanical system carefully. In this study, we designed the neural system consisting of CPGs, responses, and reflexes referring to biological concepts while taking the necessary conditions for adap- tive walking into account.
We newly employed a tonic labyrinthine reflex, a side- way stepping reflex, and a re-stepping reflex/response to make the self-contained quadruped robot (Tekken2) walk in outdoor environment. In order to increase the degrees of terrain irregularity which Tekken2 can cope with, we should employ additonal reflexes and responses, and also navigation ability at the high level using vision.
MPEG footage of these experiments can be seen at: http://www.kimura.is.uec.ac.jp.
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Hiroshi Kimura∗ Yasuhiro Fukuoka∗ Avis H. Cohen∗∗ ∗Graduate School of Information Systems
Univ. of Electro-Communications 1-5-1 Chofugaoka, Chofu Tokyo 182-8585, Japan
{hiroshi, fukuoka}@kimura.is.uec.ac.jp
Univ. of Maryland, College Park MD 20742, USA [email protected]
Abstract
We have been trying to induce a quadruped robot to walk with medium walking speed on ir- regular terrain based on biological concepts. We propose the necessary conditions for stable dy- namic walking on irregular terrain in general, and we design the mechanical system and the neu- ral system by comparing biological concepts with those necessary conditions described in physical terms. PD-controller at joints constructs the vir- tual spring-damper system as the visco-elasticity model of a muscle. The neural system model con- sists of a CPG (central pattern generator), re- flexes and responses. A CPG generates rhythmic motion for walking. A CPG receives sensory in- put and changes the period of its own active phase as responses. The virtual spring-damper system also receives sensory input and outputs torque as reflexes. The states in the virtual spring-damper system are switched based on the phase signal of the CPG. The physical oscillations such as the motion of the virtual spring-damper system of each leg and the rolling motion of the body are mutually entrained with the neural oscillations of CPGs. Consequently, the adaptive walking is generated by the interaction with environment. In this paper, we report our experimental results of dynamic walking on terrains of medium degrees of irregularity in order to verify the effectiveness of the designed neuro-mechanical system.
1. Introduction
Many previous studies of legged robots have been per- formed, including studies on running (Hodgins and Raib- ert 1991) and dynamic walking (Yamaguchi, Takanishi and Kato 1994; Kajita and Tani 1996; Chew, Pratt and Pratt 1999; Yoneda, Iiyama and Hirose 1994; Buehler et al. 1998) on irregular terrain. However, all of those studies assumed that the structure of terrain was known,
even though the height of the step or the inclination of the slope was unknown.
On the other hand, studies of autonomous dynamic adaptation allowing a robot to walk over irregular ter- rain with less knowledge of it have been started only re- cently and by only a few research groups. One example is the recent achievement of high-speed mobility of a hexa- pod over irregular terrain, with appropriate mechanical compliance of the legs (Saranli, Buehler and Koditschek 2001; Cham et al. 2002). The purpose of this study is to realize high-speed mobility on irregular terrain us- ing a mammal-like quadruped robot, the dynamic walk- ing of which is less stable than that of hexapod robots, by referring to the marvelous abilities of animals to au- tonomously adapt to their environment.
As many biological studies of motion control pro- gressed, it has become generally accepted that animals’ walking is mainly generated at the spinal cord by a combination of a CPG (central pattern generator) and reflexes receiving adjustment signals from a cerebrum, cerebellum and brain stem (Grillner 1981; Cohen and Boothe 1999). A great deal of the previous research on this attempted to generate walking using a neural system model, including studies on dynamic walking in simulation (Taga, Yamaguchi and Shimizu 1991; Taga 1995; Miyakoshi et al. 1998; Ijspeert 1998, 2001), and real robots (Ilg et al. 1999; Tsujita, Tsuchiya et al. 2001; Lewis et al. 2003). But autonomously adaptive dynamic walking on irregular terrain was rarely realized in those earlier studies except for our studies (Kimura et al. 1999, 2001; Fukuoka et al. 2003). This paper reports on our progress in the past couple of years using a newly de- veloped quadruped called “Tekken1&2,” which contains a mechanism designed for walking in 3D space (pitch, roll and yaw planes) on irregular terrain. Especially, the self-contained (power autonomous) system was realized in Tekken2 (Fig.1) and we succeeded in outdoor experi- ments using it.
Table 1: Biological concepts of legged locomotion control.
Limit-Cycle-based Control ZMP-based Control by Neural System by Mechanism
(CPG and reflexes) (spring and damper) good for posture and low medium speed walking high speed running control of speed walking
main upper neural system lower neural system musculoskeletal system controller acquired by learning (at spinal cord, brain stem, etc.) through self stabilization
battery
controller
battery
radio receiver
Figure 1: Self-contained quadruped robot: Tekken2. The
length of the body and a leg in standing are 30 [cm] and
20 [cm]. The weight including batteries is 4.3 [kg]. The hip
pitch joint, knee pitch joint and hip yaw joint are activated
by DC motors of 23 [W], 23 [W] and 8 [W] through gear ratio
of 20, 28 and 18, respectively.
2. Design concepts for adaptive walking
2.1 Legged locomotion control
Methods for legged locomotion control are classified into ZMP-based control and limit-cycle-based control (Table 1). ZMP (zero moment point) is the extension of the cen- ter of gravity considering inertia force and so on. It was shown that ZMP-based control is effective for control- ling posture and low-speed walking of a biped (Takan- ishi et al. 1990) and a quadruped (Yoneda, Iiyama and Hirose 1994). However, ZMP-based control is not good for medium or high-speed walking from the standpoint of energy consumption, since a body with a large mass needs to be accelerated and decelerated by actuators in every step cycle.
In contrast, motion generated by the limit-cycle-based control has superior energy efficiency. But there exists the upper bound of the period of the walking cycle, in which stable dynamic walking can be realized (Kimura et al. 1990). It should be noted that control by a neural system consisting of CPGs and reflexes is dominant for various kinds of adjustments in medium-speed walking of animals (Grillner 1981). Full and Koditschek (1999)
also pointed out that, in high-speed running, kinetic en- ergy is dominant, and self-stabilization by a mechanism with a spring and a damper is more important than ad- justments by the neural system. Our study is aimed at medium-speed walking controlled by neural system con- sisting of CPGs and reflexes (Table 1).
2.2 Necessary conditions for stable dynamic walking on irregular terrain
We propose the necessary conditions for stable dynamic walking on irregular terrain, which can be itemized in physical terms:
(a) the period of the walking cycle should be shorter enough than the upper bound of it, in which stable dynamic walking can be realized,
(b) the swinging legs should be free to move forward during the first period of the swing phase,
(c) the swinging legs should land reliably on the ground during the second period of the swing phase,
(d) the angular velocity of the supporting legs relative to the ground should be kept constant during their pitching motion or rolling motion around the contact points at the moment of landing or leaving,
(e) the phase difference between rolling motion of the body and pitching motion of legs should be main- tained regardless of a disturbance from irregular ter- rain, and
(f) the phase differences between the legs should be maintained regardless of delay in the pitching mo- tion of a leg receiving a disturbance from irregular terrain.
We design the neural system for these necessary condi- tions to be satisfied in order to realize adaptive walking.
2.3 Mutual entrainment between neural system and mechanical system
Motion generation based on biological concepts is illus- trated as Fig.2, where a neural system and a mechanical system have their own non-linear dynamics. The char- acteristic of this method is that there is no adaptation through motion planning. These two dynamic systems
are coupled to each other, generating motion by inter- acting with the environment emergingly and adaptively (Taga, Yamaguchi and Shimizu 1991; Taga 1995). We call this method “coupled-dynamics-based motion gen- eration.”
The coupled dynamic system can induce autonomous adaptation according to its own dynamics, under changes in the environment (e.g., adaptive walking on ir- regular terrain) and under adjustment of the neural sys- tem parameters by an upper-level controller (e.g., gait transition in change of walking speed). Therefore, we can avoid such serious problems in robotics as model- ing of mechanical system and environment, autonomous planning, conflict between planned motion and actual motion and so on.
neural system
2.4 Design of neural system
We construct the neural system centering a neural oscil- lator as a model of a CPG, since the exchange between the swing and stance phases in the short term and the quick adjustment of these phases on irregular terrain are essential in the dynamic walking of a quadruped where the unstable two-legged stance phase appears. On the other hand, Espenschied et al. (1996) constructed the gait pattern generator proposed by Cruse (1990) refer- ring to a stick insect. They also employed the sway- ing, stepping, elevator and searching reflexes observed by Pearson et al. (1984) in a stick insect. Consequently, their neural system is more sensor dependent and more decentralized by using a non-oscillator type CPG.
The characteristics of our neural system in comparison with the one used by Espenschied et al. (1996) are:
(1) The cyclic period is mainly determined by the time constant of CPGs. This makes it easy for the nec- essary condition (a) described in Section 2.2 to be satisfied.
(2) The gait is mainly determined by the connecting weights of the CPGs network.
(3) As sensor feedback for adaptation on irregular ter- rain, a “response” directly and quickly modulating
the CPG phase is employed in parallel with a “re- flex” directly generating joint torque (Fig.2).
About the characteristic (3), it is well known in physiol- ogy that
• some sensory stimuli modify CPG activity and reflex- ive responses to sensory stimuli are phase dependent under CPG activity (Cohen and Boothe 1999).
Such interaction between CPG activity and a sensory stimulus is very important for adaptation, and corre- sponds to the necessary conditions described in physical terms in Section 2.2.
2.5 Design of mechanical system
Each leg of Tekken has a hip pitch joint, a hip yaw joint, a knee pitch joint, and an ankle pitch joint (Fig.1). The direction in which Tekken walks can be changed by using the hip yaw joints. Two rate gyro sensors and two incli- nometers are mounted on the body in order to measure the body pitch and roll angles.
In order to obtain appropriate mutual entrainment be- tween neural system and mechanical system, mechanical system should be well designed to have the good dy- namic properties. In addition, performance of dynamic walking such as adaptability on irregular terrain, energy efficiency, maximum speed and so on highly depends on the mechanical design. The design concepts of Tekken are:
[a] high power actuators and small inertia moment of legs for quick motion and response,
[b] small gear reduction ratio for high backdrivability to increase passive compliance of joints,
[c] small mass of the lowest link of legs to decrease im- pact force at collision,
[d] small contacting area at toes to increase adaptability on irregular terrain,
[e] passive ankle joint mechanism to prevent a swinging leg from stumble on an obstacle quickly (Fig.3).
A
Figure 3: The ankle joint can be passively rotated:(A) if
the toe contacts with an obstacle in a swing phase, and is
locked:(B) while the leg is in a stance phase.
3. Implementation of neural system for adaptive walking
3.1 Rhythmic motion by CPG
Although actual neurons as a CPG in higher animals have not yet become well known, features of a CPG have been actively studied in biology, physiology, and so on. Several mathematical models were also proposed, and it was pointed out that a CPG has the capability to gener- ate and modulate walking patterns and to be mutually entrained with a rhythmic joint motion (Grillner 1981; Cohen and Boothe 1999; Taga, Yamaguchi and Shimizu 1991; Taga 1995). As a model of a CPG, we used a neu- ral oscillator proposed by Matsuoka (1987), and applied to the biped simulation by Taga et al. (1991; 1995). A single neural oscillator consists of two mutually inhibit- ing neurons (Fig.4-(a)). Each neuron in this model is represented by the following nonlinear differential equa- tions:
τu{e,f}i = −u{e,f}i + wfey{f,e}i − βv{e,f}i
+u0 + Feed{e,f}i + n∑
j=1
wijy{e,f}j
y{e,f}i = max (u{e,f}i, 0) (1) τ ′v{e,f}i = −v{e,f}i + y{e,f}i
where the suffix e, f , and i mean an extensor neuron, a flexor neuron, and the i-th neural oscillator, respectively. u{e,f}i is uei or ufi, that is, the inner state of an extensor neuron or a flexor neuron of the i-th neural oscillator; v{e,f}i is a variable representing the degree of the self- inhibition effect of the neuron; yei and yfi are the output of extensor and flexor neurons; u0 is an external input with a constant rate; Feed{e,f}i is a feedback signal from the robot, that is, a joint angle, angular velocity and so on; and β is a constant representing the degree of the self- inhibition influence on the inner state. The quantities τ and τ ′ are time constants of u{e,f}i and v{e,f}i; wfe is a connecting weight between flexor and extensor neurons; wij is a connecting weight between neurons of the i-th and j-th neural oscillator.
In Fig.4-(a), the output of a CPG is a phase signal: yi
. yi = −yei + yfi (2)
The positive or negative value of yi corresponds to ac- tivity of a flexor or extensor neuron, respectively.
We use the following hip joint angle feedback as a basic sensory input to a CPG called a “tonic stretch response” in all experiments of this study. This negative feedback makes a CPG be entrained with a rhythmic hip joint motion.
Feede·tsr = ktsr(θ − θ0), F eedf ·tsr = −Feede·tsr (3)
Feed{e,f} = Feed{e,f}·tsr (4)
Neural Oscillator(a)
LF : left fore leg
LH : left hind leg RF : right fore leg RH : right hind leg
(b) Neural Oscillator Network
extensor neuron of other N.O.’s
flexor neuron of other N.O.’s
Figure 4: Neural oscillator as a model of a CPG. The suffix
i, j = 1, 2, 3, 4 corresponds to LF, LH, RF, RH. L, R, F or H
means the left, right, fore or hind leg, respectively.
where θ is the measured hip joint angle, θ0 is the origin of the hip joint angle in standing and ktsr is the feedback gain. We eliminate the suffix i when we consider a single neural oscillator.
By connecting the CPG of each leg (Fig.4-(b)), CPGs are mutually entrained and oscillate in the same period and with a fixed phase difference. This mutual entrain- ment between the CPGs of the legs results in a gait. The gait is a walking pattern, and can be defined by phase differences between the legs during their pitching motion. The typical symmetric gaits are a trot and a pace. Diagonal legs and lateral legs are paired and move together in a trot gait and a pace gait, respectively. A walk gait is the transversal gait between the trot and pace gaits. We used a trot gait for most of experiments. But the autonomous gait transition in changing walking speed was discussed in our former study (Fukuoka et al. 2003).
3.2 Virtual spring-damper system
We employ the model of the muscle stiffness, which is generated by the stretch reflex and variable according to the stance/swing phases, adjusted by the neural system. The muscle stiffness is high in a stance phase for sup- porting a body against the gravity and low in a swing phase for compliance against the disturbance (Akazawa et al. 1982). All joints of Tekken are PD controlled to move to their desired angles in each of three states (A, B, C) in Fig.5 in order to generate each motion such as swinging up (A), swinging forward (B) and pulling down/back of a supporting leg (C). The timing for all joints of a leg to switch to the next state are:
A→ B: when the hip joint angle of the leg reaches the desired angle of the state (A)
B → C: when the CPG extensor neuron of the leg becomes active (yi ≤ 0)
C → A: when the CPG flexor neuron of the leg becomes active (yi > 0)
A
Figure 5: State transition in the virtual spring-damper sys-
tem. The desired joint angles in each state are shown by the
broken lines.
The desired angles and P-gain of each joint in each state are shown in Table 2, where constant values of the desired joint angles and constant P-gains were de- termined through experiments. Since Tekken has high backdrivability with small gear ratio in each joint, PD- controller can construct the virtual spring-damper sys- tem with relatively low stiffness coupled with the me- chanical system. Such compliant joints of legs can im- prove the passive adaptability on irregular terrain.
3.3 CPGs and pitching motion of legs
The diagram of the pitching motion control consisting of CPGs and the virtual spring-damper system is shown in the middle part of Fig.6. Joint torque of all joints is determined by the PD controller, corresponding to a stretch reflex at an α motor neuron in animals. The desired angle and P-gain of each joint is switched based on the phase of the CPG output: yi in Eq.(2) as described in Section 3.2. As a result of the switching of the virtual spring-damper system and the joint angle feedback signal to the CPG in Eq.(4), the CPG and the pitching motion of the leg are mutually entrained.
3.4 Reflexes and responses
It is known in physiology that the reflex to a stimu- lus on the paw dorsum in the walking of a cat depends on whether flexor or extensor muscles are active (as re- viewed in Cohen and Boothe 1999). That is,
• when flexor muscles are active, the leg is flexed in order to escape from the stimulus,
Table 2: Desired value of the joint angles and P-gains at the
joints used in the PD-controller for the virtual spring-damper
system in each state shown in Fig.5. θ, φ and ψ are the hip
pitch joint angle, the knee pitch joint angle and the hip yaw
joint angle, respectively.
P control angle in state desired value[rad] P-gain[Nm/rad]
θ in A 1.2θC→A G1
θ in B −0.17 G2v+G3
θ in C θstance+ −G4v+G5
body pitch angle
ψ in all states 0 G8
θC→A: the hip joint angle measured at the instance when the state changes from (C) to (A).
θstance: variable to change the walking speed. body pitch angle: the measured pitching angle of
the body used for the vestibulospinal reflex. ∗ means that the desired angle is calculated
on-line for the height from the toe to the hip joint to be constant.
v [m/s]: the measured walking speed of Tekken.
• when extensor muscles are active, the leg is strongly extended in order to prevent the cat from falling down.
We call these a “flexor reflex” and a “extensor reflex,” respectively, and we assume that the phase signal from the CPG of the leg switches such reflexes.
The following biological concepts are also known (Ogawa et al. 1998):
• when the vestibule in a head detects an inclination in pitch or roll plane, a downward-inclined leg is ex- tended while an upward-inclined leg is flexed (Fig.7).
We call this a “vestibulospinal reflex.”
Referring to such biological knowledge, we employed the several reflexes and responses (Fig.6) to satisfy the necessary conditions (b)∼(e) described in physical terms in Section 2.2 in addition to the stretch reflex and re- sponse described in Section 3.2 and 3.1. The necessary condition (f) can be satisfied by the mutual entrainment between CPGs and the pitching motion of legs, and the mutual entrainment among CPGs (Kimura et al. 2001).
3.4.1 Flexor and extensor reflexes
The flexor and extensor reflexes contribute to satisfy the conditions (b) and (c), respectively. In our former stud- ies (Kimura et al. 1999, 2001), the stumble of a swinging
- θ swing
α
Figure 6: Control diagram for Tekken. PD-control at the hip
yaw and knee pitch joints are eliminated in this figure.
leg on an obstacle was detected by force sensor, and the flexor or extensor reflex was activated afterwards. How- ever, the problem was that the robot typically fell down due to the delayed flexing motion caused by the delay of sensing and large inertia of the leg while walking with short cyclic period. Therefore, we substitute the flexor reflex for the passive ankle joint mechanism (Fig.3) uti- lizing the fact that the collision with a forward obstacle occurs in the first half of a swing phase. The extensor reflex has not yet been implemented in Tekken.
3.4.2 Vestibulospinal reflex and response for pitching
We call the reflex/response for an inclination in the pitch plane a “vestibulospinal reflex/response for pitching,” the role of which corresponds to the condition (d) and (f). In Tekken, hip joint torque in the stance phase is adjusted by the vestibulospinal reflex (Fig.7-(a)), since the body pitch angle is added to θstance in Table 2. For the vestibulospinal response, the following equations are used rather than Eq.(3), (4).
θvsr = θ − (body pitch angle) Feed{e,f}·tsr·vsr = ±ktsr(θvsr − θ0).
(5)
roll plane
upward -inclined
3.4.3 Vestibulospinal reflex and response for rolling
We call the reflex/response for an inclination in the roll plane a “tonic labyrinthine reflex/response1 .”
• Tonic labyrinthine reflex
The tonic labyrinthine reflex (TLRF) is employed as the adjustment of P-gain of the knee joint of the sup- porting legs (G7 in Table 2) according to Eq.(7).
G7 = δ(leg) ktlrf × (body roll angle) +G7 (7)
δ(leg) = {
1, if leg is a right leg; −1, otherwise
When an inclination of a body in roll plane (body roll an- gle) is detected, the knee joint P-gain of the downward- inclined legs is increased to extend those legs. In addi- tion, the knee joint P-gain of the upward-inclined legs is decreased to flex those lges. As a result, the inclination of the body in roll plane is decreased (Fig.7-(b)).
• Tonic labyrinthine response
On the other hand, the tonic labyrinthine response (TLRS) is employed as rolling motion feedback to CPGs (upper left part of Fig.6). In Tekken, we made the body roll angle be inputted to the CPGs as a feedback signal expressed by Eq.(8), (9).
Feede·tlrs = δ(leg) ktlrs × (body roll angle) Feedf ·tlrs = −Feede·tlrs
(8)
Feedf = Feedf ·tsr·vsr + Feedf ·tlrs (9)
1The “tonic labyrinthine reflex” is defined in Ogawa et al. (1998). The same reflex is called “vestibular reflex” in Ghez (1991).
The rolling motion feedback to CPGs: Eq.(8), (9) con- tributes to an appropriate adjustment of the periods of the stance and swing phases while walking on irregular terrain (Fukuoka et al. 2003).
The rolling motion feedback to CPGs is important also in order to satisfy the condition (e). CPGs, the pitch- ing motions of the legs, and the rolling motion of the body are mutually entrained through the rolling mo- tion feedback to CPGs (Fig.8). This means that the rolling motion can be the standard oscillation for whole oscillations, in order to compensate for the weak connec- tion between the fore and hind legs in the CPG network (Fig.4-(b)). As a result, the phase difference between the fore and hind legs is fixed, and the gait becomes stable.
Excitatory Connection Inhibitory Connection
angle
Figure 8: Relation among CPGs, pitching motion of a leg
and rolling motion of a body
3.4.4 Stepping reflex to stabilize rolling motion
It is known that the adjustment of the sideway touch- down angle of a swinging leg is effective in stabiliz- ing rolling motion against disturbances (Miura and Shi- moyama 1994). We call this a “sideway stepping reflex,” which helps to satisfy the condition (d) during rolling motion. The sideway stepping reflex is effective also in walking on a sideway inclined slope.
For examples, when Tekken walks on a right-inclined slope (Fig.9), Tekken continues to walk while keeping the phase differences between left and right lges with the help of the tonic labyrinthine response. But Tekken can- not walk straight and shifts its walking direction to the right due to the difference of the gravity load between left and right legs. In addition, Tekken typically falls down to the right for the perturbation from the left in the case of Fig.9-(a), since the wide stability margin: WSM 2 is small. The sideway stepping reflex helps to stabilize the
2the shortest distance from the projected point of the center of gravity to the edges of the polygon constructed by the projected points of legs independent of their stance or swing phases (Fukuoka et al. 2003)
walking direction and to prevent the robot from falling down while keepingWSM large on such sideway inclined slope (Fig.9-(b)).
front view
WSM WSM
(a) (b)
a sideway stepping reflex, (b):With.
Since Tekken has no joint round the roll axis, the side- way stepping reflex is implemented as changing the de- sired angle of the hip yaw joint from 0 (Table 2) to ψ∗
according to Eq.(10).
ψ∗ = δ(leg) kstpr × (body roll angle) (10)
3.4.5 Stepping reflex to stabilize forward speed
In Table 2, P-gain of hip joints in the state (B) and (C) is adjusted using the measured walking speed: v. For examples, when v is increased in the state (B) as the swing phase, the swinging legs land on the forwarder position because P-gain is increased and the swinging legs are more strongly pulled forward. As a result, the increase of v is depressed since the forward motion of the inverted pendulum in the next stance phase is depressed by the gravity (Miura and Shimoyama 1994). On the other hand, when v is decreased in the state (C) as the stance phase while walking up a slope, the increase of P-gain generates additional torque against the gravity load at hip joints of the supporting legs. As a result, the decrease of v is depressed. These reflexes contribute to satisfy the condition (d).
3.4.6 Re-stepping reflex and response for walk- ing down a step
When loss of ground contact is detected in a swing phase while walking over a ditch, a cat activates re-stepping to extend the swing phase and make the leg land on the forwarder position (Hiebert et al. 1994). We call this “re-stepping reflex/response,” which is effective for the necessary condition (c) and (d) to be satisfied also in walking down a large step (Fig.10).
re-stepping
4. Experiments
4.1 Experiments under instantaneous distur- bances
We made Tekken1 walk on several terrains of medium de- gree of irregularity in indoor environment (Fig.11) with all responses and five reflexes, those are, the stretch reflex, flexor reflex, vestibulospinal reflex for pitching, stepping reflex for forward speed and re-stepping re- flex, among seven reflexes described in Section 3.4. This means that Tekken can cope with the instantaneous dis- turbances while walking over irregular terrain consist- ing of series of steps and short slopes. Without a tonic labyrinthine response, the gait was greatly disturbed on irregular terrain, even if Tekken1 didn’t fall down.
(a) (b)
(c) (d)
Figure 11: Walking over a step 4 [cm] in height: (a), walking
up and down a slope of 10 [deg] in a forward direction: (b),
walking over pebbles: (c) and walking over sideway slopes of
3 and 5 [deg]: (d)
Especially noteworthy, Tekken1 successfully walked down a large step with approx. 0.5 [m/s] speed using the re-stepping reflex/response (Fig.12). Without the re-stepping reflex/response, Tekken1 typically fell down
6.0 6.5 7.0 7.55.5
flexor
extensor
0.14[s]
re-stepping response
Figure 12: Walking down a step of 7 [cm] in height. A re-
stepping response was activated when the contact of the right
fore leg had not been detected for 0.14 [s] after the activity
of the flexor neuron became zero.
forward because fore legs landed on the backwarder po- sition excessively and could not depress the increased forward speed.
4.2 Experiments under long-lasting distur- bances
We made Tekken2 walk on a right-inclined slope of 4 [deg] (0.07 [rad]) in indoor environment with all re- sponses and reflexes described in Section 3.4 in order to confirm the effectiveness of a tonic labyrinthine re- flex and a sideway stepping reflex (Fig.9-(b)) under long- lasting disturbances. As a result of the experiment, the body roll angle and the hip yaw angle ψ of the right hindleg and left hindleg are shown in Fig.13. Tekken2 had walked on the right-inclined slope for 2 [sec], and walked on flat terrain afterwards.
In Fig.13, we can see that the body roll angle was posi- tive (0.05∼0.14 [rad]) while walking on the right-inclined slope. The hip yaw joint of the right hindleg moved to the outside of the body (right) by approx. 0.04 [rad] due to the sideway stepping reflex in the swing phase and moved to the inside of the body by approx. -0.04 [rad] due to the gravity load in the stance phase. On the other hand, the hip yaw joint of the left hindleg moved to the inside of the body (right) by approx. -0.04 [rad] due to the sideway stepping reflex in the swing phase and moved to the inside of the body furthermore by approx. -0.1 [rad] due to the gravity load in the stance phase. Consequently, Tekken2 succeeded in straight walking on the sideway inclined slope.
4.3 Outdoor experiments using Tekken2
Even on a paved road in outdoor environment, there ex- ist a slope of 3 [deg] at most, bumps of 1 [cm] in height and small pebbles everywhere. With all responses and reflexes described in Section 3.4, Tekken2 successfully maintained a stable gait on the paved road for 60 [sec]
-0.1
-0.05
0.05
0.1
0.15
[rad]
[sec]
body roll angle ψ of right hindleg ψ of left hindleg
inclination of a slope (0.07 [rad])
swing phase
stance phase
swing phase
stance phase
Figure 13: Walking on a right-inclined slope of 4 [deg]
Figure 14: Photos of walking in outdoor environment
with approx. 0.5 [m/s] speed while changing its walk- ing speed and direction by receiving the operation com- mands from the radio controller (Fig.14).
5. Discussion : how to determine the val- ues of parameters
Values of all parameters in the neural system including the virtual spring-damper system except for θstance were determined experimentally. But it should be noted that those values in each robot were fixed in all experiments independent of kinds of terrain. Although the size and weight of Tekken2 are different from those of Tekken1, the values of the parameters of CPGs used for Tekken2 were same with those used for Tekken1.
The most important subject in coupled-dynamics- based motion generation is to design and construct a neural system carefully, while taking into account the dy- namics of a mechanical system and its interaction with the environment. In this study, we designed a neural system consisting of CPGs, responses and reflexes, while taking into account the characteristics of dynamic walk- ing and utilizing knowledge and concepts in physics, bi- ology, physiology and so on described in Section 2. and 3.. The relationship between parameters of CPGs and
the mechanical system was previously analyzed to some extent (Kimura et al. 2001). Since the relationship be- tween parameters of reflexes/responses and the mechan- ical system has not yet been investigated, we manually determined values of those parameters through experi- ments. The issue of how to construct a neural system suitable for a mechanical system corresponds to the is- sue of “embodiment” at the lowest level of sensorimotor coordination.
In addition, we pointed out the trade-off problem be- tween the stability and the energy consumption in deter- mining the cyclic period of walking on irregular terrain, and showed one example to solve this problem (Fukuoka et al. 2003).
6. Conclusion
In the neural system model proposed in this study, the relationships among CPGs, sensory input, reflexes and the mechanical system are simply defined, and motion generation and adaptation are emergingly induced by the coupled dynamics of a neural system and a mechanical system by interacting with the environment. To gener- ate appropriate adaptation, it is necessary to design both the neural system and the mechanical system carefully. In this study, we designed the neural system consisting of CPGs, responses, and reflexes referring to biological concepts while taking the necessary conditions for adap- tive walking into account.
We newly employed a tonic labyrinthine reflex, a side- way stepping reflex, and a re-stepping reflex/response to make the self-contained quadruped robot (Tekken2) walk in outdoor environment. In order to increase the degrees of terrain irregularity which Tekken2 can cope with, we should employ additonal reflexes and responses, and also navigation ability at the high level using vision.
MPEG footage of these experiments can be seen at: http://www.kimura.is.uec.ac.jp.
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