review: differential kinematics find the relationship between the joint velocities and the...

Review: Differential Kinematics Review: Differential Kinematics Find the relationship between the joint velocities a

nd the end-effector linear and angular velocities.

Linear velocity

Angular velocity

i

ii dq

for a revolute joint

for a prismatic joint

Review: Differential Kinematics Review: Differential Kinematics Approach 1

q

qpJ P

)(


Prismatic joint

Revolute joint

nOn

Pni

Oi

Pi

O

P qJ

Jq

J

Jq

J

Jv

1

1

1

The contribution of single joint i to the end-effector angular velocity

The contribution of single joint i to the end-effector linear velocity


Kinematic SingularitiesKinematic Singularities

The Jacobian is, in general, a function of the configuration q; those configurations at which J is rank-deficient are termed Kinematic singularities.

Reasons to Find SingularitiesReasons to Find Singularities

Singularities represent configurations at which mobility of the structure is reduced

Infinite solutions to the inverse kinematics problem may exist

In the neighborhood of a singularity, small velocities in the operational space may cause large velocities in the joint space

Problems near Singular PositionsProblems near Singular Positions

The robot is physically limited from unusually high joint velocities by motor power constraints, etc. So the robot will be unable to track this joint velocity trajectory exactly, resulting in some perturbation to the commanded cartesian velocity trajectory

The high accelerations that come from approaching too close to a singularity have caused the destruction of many robot gears and shafts over the years.

Classification of SingularitiesClassification of Singularities

Boundary singularities that occur when the manipulator is either outstretched or retracted. Not true drawback

Internal singularities that occur inside the reachable workspace Can cause serious problems

Example 3.2: Two-link Planar ArmExample 3.2: Two-link Planar Arm Consider only planar components of linear velocity

Consider determinant of J

Conditions for singularity

Example 3.2: Two-link Planar ArmExample 3.2: Two-link Planar Arm Conditions for sigularity

Jacobian when theta2=0

12121

12121

)(

)(

cacaa

sasaaJ

Computation of internal singularity via the Jacob

ian determinant

Decoupling of singularity computation in the cas

e of spherical wrist

Wrist singularity

Arm singularity

Singularity DecouplingSingularity Decoupling


Wrist Singularity Z3, z4 and z5 are linearly dependent

Cannot rotate about the axis

orthogonal to z4 and z3


Elbow Singularity Similar to two-link planar arm

The elbow is outstretched or retracted


Arm Singularity

The whole z0 axis describes a continuum of singular configurations

0

0023322

y

x

p

pcaca


Arm Singularity A rotation of theta1 does not cause

any translation of the wrist position The first column of JP1=0

Infinite solution

Cannot move along the z1 direction The last two columns of JP1 are

orthogonal to z1

Well identified in operational space; Can be suitably avoided in the path

planning stage

Differential Kinematics InversionDifferential Kinematics Inversion

Inverse kinematics problem: there is no general purpose technique Multiple solutions may exist Infinite solutions may exist There might be no admissible solutions

Numerical solution technique in general do not allow computation of all admissible

solutions


Suppose that a motion trajectory is assigned to the end effector in terms of v and the initial conditions on position and orientations

The aim is to determine a feasible joint trajectory (q(t), q’(t)) that reproduces the given trajectory

Should inverse kinematics problems be solved?


Solution procedure:

If J is not square? (redundant)

If J is singular?

If J is near singularity?

Analytical JacobianAnalytical Jacobian

The geometric Jacobian is computed by following a geometric technique

Question: if the end effector position and orientation are specified in terms of minimal representation, is it possible to compute Jacobian via differentiation of the direct kinematics function?


Analytical technique


Analytical Jacobian

For the Euler angles ZYZ


From a physical viewpoint, the meaning of ώ is more intuitive than that of φ’

On the other hand, while the integral of φ’ over time gives φ, the integral of ώ does not admit a clear physical interpretation

Example 3.3Example 3.3

StaticsStatics

Determine the relationship between the generalized forces applied to the end-effector and the generalized forces applied to the joints - forces for prismatic joints, torques for revolute joints - with the manipulator at an equilibrium configuration.

X0

Y0

x0

y0

0

Y1X1

0

x2

a1

v

vv

v

R

a2

y2

fx

fy

Let τ denote the (n×1) vector of joint torques and γ(r ×1) vector of end effector forces (exerted on the environment) where r is the dimension of the operational space of interest

StaticsStatics

)(qJ T

X0

Y0

x0

y0

0

Y1X1

0

x2

a1

v

vv

v

R

a2

y2

fx

fy

Manipulability EllipsoidsManipulability Ellipsoids

Velocity manipulability ellipsoid Capability of a manipulator to arbitrarily change the en

d effector position and orientation


Velocity manipulability ellipsoid Manipulability measure: distance of the manipulator fr

om singular configurations

Example 3.6


Force manipulability ellipsoid


Manipulability ellipsoid can be used to analyze compatibility of a structure to execute a task assigned along a direction Actuation task of velocity (force) Control task of velocity (force)


Control task of velocity (force) Fine control of the vertical force Fine control of the horizontal velocity


Actuation task of velocity (force) Actuate a large vertical force (to

sustain the weight) Actuate a large horizontal velocity

review: differential kinematics find the relationship between the joint velocities and the...

Documents

high joint velocities

end effector

angular velocities

joint spaceproblems

joint velocity trajectory

small velocities

large velocities

link planar armconsider