Review: Differential Kinematics Find the relationship between the joint velocities and the...
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Transcript of 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
)(
Review: Differential Kinematics Review: Differential Kinematics Approach 2
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
Review: Differential Kinematics Review: Differential Kinematics Approach 3
Review: Differential Kinematics Review: Differential Kinematics Approach 3
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
Singularity DecouplingSingularity Decoupling
Wrist Singularity Z3, z4 and z5 are linearly dependent
Cannot rotate about the axis
orthogonal to z4 and z3
Singularity DecouplingSingularity Decoupling
Elbow Singularity Similar to two-link planar arm
The elbow is outstretched or retracted
Singularity DecouplingSingularity Decoupling
Arm Singularity
The whole z0 axis describes a continuum of singular configurations
0
0023322
y
x
p
pcaca
Singularity DecouplingSingularity Decoupling
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
Differential Kinematics InversionDifferential Kinematics Inversion
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?
Differential Kinematics InversionDifferential Kinematics Inversion
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 JacobianAnalytical Jacobian
Analytical technique
Analytical JacobianAnalytical Jacobian
Analytical Jacobian
For the Euler angles ZYZ
Analytical JacobianAnalytical Jacobian
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
Manipulability EllipsoidsManipulability Ellipsoids
Velocity manipulability ellipsoid Manipulability measure: distance of the manipulator fr
om singular configurations
Example 3.6
Manipulability EllipsoidsManipulability Ellipsoids
Force manipulability ellipsoid
Manipulability EllipsoidsManipulability Ellipsoids
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)
Manipulability EllipsoidsManipulability Ellipsoids
Control task of velocity (force) Fine control of the vertical force Fine control of the horizontal velocity
Manipulability EllipsoidsManipulability Ellipsoids
Actuation task of velocity (force) Actuate a large vertical force (to
sustain the weight) Actuate a large horizontal velocity