hybrid operational space control for compliant quadruped
TRANSCRIPT
Hybrid Operational Space Control for Compliant
Quadruped Robots
Marco Hutter and Roland SiegwartASL, ETH Zurich, Tannenstrasse 3, CLA E32, 8092 Zurich, Switzerland
Keywords - Compliant Design, Quadrupedal Lo-comotion, Hybrid Operational Space Control
Abstract - This work introduces the concept ofhybrid operational space control, a method thatunifies kinematic tracking of individual joints ofa robot with an inverse dynamics task space con-troller for its remainder. The proposed controlstrategy allows for a hierarchical task decompo-sition while simultaneously regulating the innerforces between the contact points. At the sametime it improves fast tracking for compliant sys-tems by means of appropriate low level positioncontrollers. Introducing StarlETH, a compliantquadrupedal robot, the applicability of the con-troller and the hardware is demonstrated in re-altime simulations and hardware experiments.We perform static walking in challenging ter-rain and show how the controller can combineprecise and fast position control with robust andcompliant interaction with the environment.
1 Introduction
The dynamic walking community seeks for hardwaredevices that are compliant and torque controllable. Thefirst requirement is motivated by safety and efficiencyconsiderations: Legged systems should interact softlywith their environment to ensure robustness againstdisturbances such as terrain irregularities or slippage,to protect their hardware from damage through unex-pected collisions, and to safely work hand-in-hand withhuman collaborators. At the same time - and this issimilar to muscles and tendons found in nature -, theyshould be able to use their passive system complianceto locomote fast and efficiently through periodicallystoring and releasing energy. The second requirement,full torque controllability, is needed to make the sys-tem highly versatile. Different control concepts (e.g.[6],[2], [3],...) based on inverse dynamics methods haveshown impressive performance in terms of executinghighly complex maneuvers or compliantly interactingwith the environment while simultaneously optimizingthe contact force distribution [5]. At the same time -and again similar to nature -, torque control enablesthe natural dynamics of pendulum motion.
motor+gearbox=
velocity source
torquesensor
low-levelcontrol
torquecontroller
modelbased
tracking
desT ,des desIj U
measT
measT
measj
desj ,j j&
Figure 1: Position control of compliant joints (SEA) thatare considered as ”torque controllable” signif-icantly limits the performance due to the cas-caded structure.
Unifying the requirements of compliance and torquecontrollability in one single legged device, we developedthe quadruped robot StarlETH that was presented tothe community at DW2011. Using high compliant se-ries elastic actuators, this system is perfectly suited fordynamic maneuvers. The springs protect the motor andgearbox, allow for energy storage during stance phase,and provide means for precise torque control. As a sub-stantial drawback, the integration of compliance in thesystem is accompanied by a loss of torque control band-width. While this is not an issue for compliant interac-tion with the environment, it greatly limits the perfor-mance in terms of fast and accurate position trackingcontrol - skills that are indispensable for foot or handpositioning while walking or manipulating. The prob-lematic is obvious: Control approaches that considerthe actuator as torque sources translate position track-ing tasks into desired joint torques. This results in acascaded structure that lowers the bandwidth in everyloop.
We are approaching this problem by considering themotor as a velocity source. A position controller is im-plemented on joint level as an LQG control structurethat includes the series-elasticity model. This setupperforms significantly better in fast and precise track-ing tasks [1] than a cascaded structure as depicted inFig. 1. Using a combination of low level position andtorque controllers, we present the method of hybrid op-erational space control: In this framework, all dynamictasks are brought into a prioritized task space controlstructure (similar to [7]) while certain joints are po-sition controlled based on traditional inverse kinemat-ics control methods. This framework measures and es-timates the influence of the position controlled jointsonto the torque controlled parts such that we can com-pensate for these effects and achieve exactly the same
task space behavior as with a purely torque controlledsystem, while at the same time tracking performance ofthe kinematically controlled segments can be improved.Several cases such as legged locomotion or robot-humaninteraction fit perfectly into this scenario. In this workwe focus only on quadrupedal walking using StarlETHin simulations and for experimental validation. The fo-cus thereby is put on locomotion in very rough terrainthat requires precise foot tracking, compliant Centerof Gravity (CoG) control, and contact force regulationwith respect to changing surface normals to avoid slip-page.
2 Inverse Dynamics with Position Control andInternal Forces
The equations of motion for a floating base system are
Mq̈ + b + g + JTs Fs = ST τ . (1)
Since the contact constraints r̈s = Jsq̈+ J̇sq̇ = 0 needsto be fulfilled, a support null-space projection allows toget rid of the contact force [4]:
P (Mq̈ + b + g) = PST τ (2)
PJTs = 0 ∀q (3)
Using the description (2), inverse dynamics can be usedto solve for the actuator torque τm required to generatea certain motion
τm =(PST
)+P (Mq̈ + b + g) . (4)
2.1 Hierarchical Task DecompositionIn our framework, the desired joint space accelerationq̈ is obtained by a hierarchical task decomposition,whereby each individual task is described as
r̈i = Jiq̈ + J̇iq̇. (5)
Tasks with lower priority are projected into the nullspace N = N (J) of higher prioritized tasks:
q̈ =
n∑j=1
Nj−1q̈j (6)
Using (5) and (6) allows calculating the joint accelera-tion q̈i for the ith task by
q̈i = (JiNi−1)+
r̈i − J̇iq̇− Ji
i−1∑j=1
Nj−1q̈j
. (7)
2.2 Hybrid OSCTo step from a purely torque controlled system to acombination of low level position and torque controllers,we separate the dynamics using the selection matricesSp and St:
q̈ = STp q̈p + ST
t q̈t (8)
Instead of demand torque signals, the kinematicallycontrolled joints qp take position and velocity com-mands as low level control input. To compensate fortheir influence on the rest of the system, we need to esti-mate the joint acceleration q̈p = ˆ̈qp. Since this is hardlymeasurable (sensor noise when joint encoder values aredouble differentiated), we rely on measured torques τ̂ p
in the corresponding joints. After some matrix algebra,the accelerations of the position controlled joints canbe estimated as
ˆ̈qp =(SpMST
p
)−1 (τ̂ p − SpMST
t q̈t − Sp (b + g)).(9)
2.3 Contact Force OptimizationIn addition to generating a desired motion, this frame-work allows also to modulate the contact force of thismulti-contact system without influencing q̈ through thenull-space of PST (4):
τ = τm +N(PST
)τ 0 = τm + NP τ 0 (10)
τ 0 allows to change the total contact force about
Fs0 =(JTs
)+STNP τ 0 = Aτ 0. (11)
In most cases, this additional contact force is requiredto minimize a certain cost function that can be statedin the form
minimizeτ0
‖Fdes −D (Fsm + Aτ 0)‖2 . (12)
In this formulation, D can be used as a selection ma-trix to load or unload specific legs before/after lift-offsuch that Fdes = DFs can be freely chosen. Hence,at the same time while shifting the CoG the leg thatjust made contact can be continuously loaded and thesuccessive swing leg can be unloaded. This ensures avery smooth transition between different support stateswhile walking.
Similar to that, using D as a projection matrix alsoallows to increase walking robustness through align-ing the contact forces with the corresponding surfacenormal direction, respectively to minimize the contactforces projected into the tangential contact plane. Inboth cases, the analytical optimal solution can be foundusing a pseudo-inverse
τ 0 = (DA)+
(Fdes −DFsm) . (13)
As long as DA has full rank, the desired force is achiev-able, otherwise this returns the least-square optimalvalue. To minimize slippage, D is set to select the tan-gential contact directions with the corresponding de-sired force Fdes equal to zero.
Figure 2: Experiments with StarlETH on a tree stem.
3 Simulation and Experimental Results
The presented structure was tested in walking simu-lations and experiments with the quadrupedal robotStarlETH. As a first outcome, we could show that thehybrid OSC framework allows position control in theswing leg joints and simultaneous torque control in thesupport leg joints. It combines advantages of bothstrategies and is a solution to the bandwidth problemof SEAs. In simulations we were able to approve thathybrid OSC is exactly equal to standard inverse dynam-ics in case of perfect joint torque sensing. Since leggedsystems usually have a very lightweight leg structure(end-effector), the controller is robust even in case oflarge torque measurement errors of τ̂ p in the range of100%. In addition to the simulation results, we alsodemonstrated the applicably in actual experiments.
As a second contribution, both aspects like smooth con-tact force distribution as well as tangential force mini-mization were tested based on the least-square optimalsolution (13) as well as through constraint friction min-imization that is implemented using an SQP solver. Itturns out that the least square optimization performsnearly as good as the complex optimization in termsof robustness against slipping but requires much lesscomputational power.
We are currently extending these principles to morechallenging terrains. Inspired by the Japanese TV showTakeshi’s Castle, we perform tests such as walking ona tree stem (Fig. 2) that require (i) compliant CoGcontrol, (ii) precise and fast foot point tracking, and(iii) modulation of the internal forces.
4 Conclusion & Open questions
This work presents two important aspects of inversedynamics control with compliant walking machines.Firstly, a hybrid operational space control implementa-tion with low level position control at the swing leg and
torque control at the stance leg joints solves the band-width problematic of series elastic systems. Second, thecontact force optimization of specific directions highlyimproved system robustness against slippage and at thesame time allowed a smooth contact force distributionamong the support legs.
We would like to discuss our results in detail withpeople that were working with similar control frame-works (whole body control, inverse dynamics, etc.).Since most researcher in this field are more focused onhigh-level control using existing hardware platforms, itwould be great to share their experience and problems.Is a hybrid task decomposition necessary? What areour/their performance limitations? How are other peo-ple dealing with passive system compliance? Is compli-ance really the right way to go or do we have to buildstiff systems and control them soft to get rid of thebandwidth problematic? What are the requirements ofthe community for novel types of actuators?
References
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