ME 537 - Robotics ME 537 - Robotics ME 537 - Robotics ME 537 - Robotics DIFFERENTIAL KINEMATICS Purpose: The purpose of this chapter is to introduce you to robot motion. Differential forms of the homogeneous transformation can be used to examine the pose velocities of frames. This method will be compared to a conventional dynamics vector approach. The Jacobian is used to map motion between joint and Cartesian space, an essential operation when curvilinear robot motion is required in applications such as welding or assembly.
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ME 537 - Robotics DIFFERENTIAL KINEMATICS Purpose: The purpose of this chapter is to introduce you to robot motion. Differential forms of the homogeneous.
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The purpose of this chapter is to introduce you to robot motion. Differential forms of the homogeneous transformation can be used to examine the pose velocities of frames. This method will be compared to a conventional dynamics vector approach. The Jacobian is used to map motion between joint and Cartesian space, an essential operation when curvilinear robot motion is required in applications such as welding or assembly.
Purpose:
The purpose of this chapter is to introduce you to robot motion. Differential forms of the homogeneous transformation can be used to examine the pose velocities of frames. This method will be compared to a conventional dynamics vector approach. The Jacobian is used to map motion between joint and Cartesian space, an essential operation when curvilinear robot motion is required in applications such as welding or assembly.
Now if we have a frame T relating a set of axes (primed axes) to global or base axes, then a small differential displacement of these axes (dp, d = dk) relative to the base axes results in a new frame T' = T + dT = H(ddp) T. Although finite rotations can't be considered vectors, differential (infinitesimal) rotations can, thus d = d k. Thus, the form of dT can be expressed as
dT = (H(ddp) - I) T = T (in base frame)
Now if we have a frame T relating a set of axes (primed axes) to global or base axes, then a small differential displacement of these axes (dp, d = dk) relative to the base axes results in a new frame T' = T + dT = H(ddp) T. Although finite rotations can't be considered vectors, differential (infinitesimal) rotations can, thus d = d k. Thus, the form of dT can be expressed as
In the case of pure rotation, the differential transformation, depicted by (or T ), reduces to a screw rotation about the unit vector k. Taking the derivative of the matrix we get the screw (angular) velocity matrix:
In the case of pure rotation, the differential transformation, depicted by (or T ), reduces to a screw rotation about the unit vector k. Taking the derivative of the matrix we get the screw (angular) velocity matrix:
Vector w can be resolved into components p in the base frame by the equation p = T w. Now under a small displacement, frame T can be expressed by the new frame
T' = T + dT = T + ∆T
A vector w in frame T, once perturbed, can be located globally by p' such that
p' = (T + T) w
The delta move, p' - p, becomes
p' - p = (T + T) w - T w = T w
Vector w can be resolved into components p in the base frame by the equation p = T w. Now under a small displacement, frame T can be expressed by the new frame
T' = T + dT = T + ∆T
A vector w in frame T, once perturbed, can be located globally by p' such that
A point P is located by vector in an offset frame C as shown. At this instant the frame C origin is being translated by the velocity vt
= 2 I + 2J m/s relative to the base frame. The frame is also being rotated relative to the base frame by the angular velocity = 2 rad/s K where I, J, K are the unit vectors for the base frame.
A point P is located by vector in an offset frame C as shown. At this instant the frame C origin is being translated by the velocity vt
= 2 I + 2J m/s relative to the base frame. The frame is also being rotated relative to the base frame by the angular velocity = 2 rad/s K where I, J, K are the unit vectors for the base frame.
Determine the instantaneous velocity of point P using both the conventional vector dynamics approach and the differential methods in this chapter.
Jacobians - serial robots Jacobians - serial robots
The Jacobian is a mapping of differential changes in one space to another space. Obviously, this is useful in robotics because we are interested in the relationship between Cartesian velocities and the robot joint rates.
The Jacobian is a mapping of differential changes in one space to another space. Obviously, this is useful in robotics because we are interested in the relationship between Cartesian velocities and the robot joint rates.
Jacobians - serial robots Jacobians - serial robots
Waldron in the paper "A Study of the Jacobian Matrix of Serial Manipulators", June, 1985, ASME Journal of Mechanisms, Transmissions, and Automation in Design, determines simple recursive forms of the Jacobian, given points on the joint axes (ri) and joint direction vectors (ei) for the joint axes
Waldron in the paper "A Study of the Jacobian Matrix of Serial Manipulators", June, 1985, ASME Journal of Mechanisms, Transmissions, and Automation in Design, determines simple recursive forms of the Jacobian, given points on the joint axes (ri) and joint direction vectors (ei) for the joint axes
Jacobians - serial robots Jacobians - serial robots
Given the Jacobian, what are we really interested in calculating for tasks like seam welding, painting, etc.?Given the Jacobian, what are we really interested in calculating for tasks like seam welding, painting, etc.?
qq == JJ-1-1 V V
Note that J-1 only exists for a six-axis robot. Robots with less than 6 joints map the joint space into a subspace of Cartesian space.
Note that J-1 only exists for a six-axis robot. Robots with less than 6 joints map the joint space into a subspace of Cartesian space.
Nearness to SingularitiesNearness to Singularities
The Jacobian determinant can always be expanded into the form
|J| = s(q) = s1(q) s2(q)…sk(q)
If one or more si(q) = 0, then we are at one or more singular configurations. If any si(q) are small then we are near singular configuration(s). Joint rates will increase near singularities.
The Jacobian determinant can always be expanded into the form
|J| = s(q) = s1(q) s2(q)…sk(q)
If one or more si(q) = 0, then we are at one or more singular configurations. If any si(q) are small then we are near singular configuration(s). Joint rates will increase near singularities.
Nearness to SingularitiesNearness to Singularities
The Jacobian can also be decomposed into the following form by Singular Value Decomposition (SVD)
J = U S VT
where U is a matrix of basis vectors in Cartesian Space, VT is a matrix of basis vectors in joint space, and S is a diagonal matrix of singular values that decrease down the diagonal. When the diagonal singular values approach zero the mechanism approaches singular configurations. Methods for computing SVD are computationally expensive.
The Jacobian can also be decomposed into the following form by Singular Value Decomposition (SVD)
J = U S VT
where U is a matrix of basis vectors in Cartesian Space, VT is a matrix of basis vectors in joint space, and S is a diagonal matrix of singular values that decrease down the diagonal. When the diagonal singular values approach zero the mechanism approaches singular configurations. Methods for computing SVD are computationally expensive.
Differential motion summary Differential motion summary 1. Differential forms can calculate velocities of frames and be simpler to
apply than the conventional vector equations.
2. Jacobian is important in robotics because the robot is controlled in joint space, whereas the robot tool is applied in physical Cartesian space. Unfortunately, motion in Cartesian space must be mapped to motion in joint space through an inverse Jacobian relation. This relation might be difficult to obtain and many robots have singular configurations that must be avoided.
3. The equations for the Jacobian are easy to determine for a robot, given the current robot configuration, as outlined by Waldron et. al in the paper "A Study of the Jacobian Matrix of Serial Manipulators", June, 1985, ASME Journal of Mechanisms, Transmissions, and Automation in Design.
1. Differential forms can calculate velocities of frames and be simpler to apply than the conventional vector equations.
2. Jacobian is important in robotics because the robot is controlled in joint space, whereas the robot tool is applied in physical Cartesian space. Unfortunately, motion in Cartesian space must be mapped to motion in joint space through an inverse Jacobian relation. This relation might be difficult to obtain and many robots have singular configurations that must be avoided.
3. The equations for the Jacobian are easy to determine for a robot, given the current robot configuration, as outlined by Waldron et. al in the paper "A Study of the Jacobian Matrix of Serial Manipulators", June, 1985, ASME Journal of Mechanisms, Transmissions, and Automation in Design.
Jacobians - parallel robots Jacobians - parallel robots Let us define two vectors:
x is a vector that locates the moving platformq is a vector of the n actuator joint values.
In general the number of actuated joints is equal to the degrees-of-freedom of the robot. The kinematics constraints imposed by the limbs will lead to a series of n constraint equations:
f(x,q) = 0 Eqn (4.42) in notes
Let us define two vectors:
x is a vector that locates the moving platformq is a vector of the n actuator joint values.
In general the number of actuated joints is equal to the degrees-of-freedom of the robot. The kinematics constraints imposed by the limbs will lead to a series of n constraint equations:
Jacobians - parallel robots Jacobians - parallel robots We can differentiate (4.42) with respect to time to generate a relationship between joint rates and moving table velocities:
where Jx = f/x and Jq = - f/q. Thus, the Jacobian equation
can be written in the form
where J = Jq-1 Jx. Note that the equation is already in the
desired inverse form, given that we can determine Jq-1 and Jx.
We can differentiate (4.42) with respect to time to generate a relationship between joint rates and moving table velocities:
where Jx = f/x and Jq = - f/q. Thus, the Jacobian equation
can be written in the form
where J = Jq-1 Jx. Note that the equation is already in the
desired inverse form, given that we can determine Jq-1 and Jx.
Singularity conditions - parallel robots Singularity conditions - parallel robots A parallel manipulator is singular when either Jq or Jx or
both are singular. If det(Jq) = 0, we refer to the singularity
as an inverse kinematics singularity. This is the kind of singularity where motion of the actuation joints causes no motion of the platform along certain directions. For the picker robot this occurs whenthe limb links all lie in a plane (2i = 0 or ) or when upper
arm links are colinear (3i = 0 or ).
A parallel manipulator is singular when either Jq or Jx or
both are singular. If det(Jq) = 0, we refer to the singularity
as an inverse kinematics singularity. This is the kind of singularity where motion of the actuation joints causes no motion of the platform along certain directions. For the picker robot this occurs whenthe limb links all lie in a plane (2i = 0 or ) or when upper
singularity as a direct kinematics singularity. This case occurs for the picker robot when all upper arm linkages are in the plane of the moving platform. Note that the actuators cannot resist any force applied to the moving platform in the z direction
If det(Jx) = 0, we refer to the
singularity as a direct kinematics singularity. This case occurs for the picker robot when all upper arm linkages are in the plane of the moving platform. Note that the actuators cannot resist any force applied to the moving platform in the z direction
1. The Jacobian in robotics is particularly important because it represents the rate mapping between joint and Cartesian space. Unfortunately, for serial robots the desire is to determine the joint rates to get desired motion in Cartesian space as the tool is moved along some line in space. This usually requires the inverse of the Jacobian, which may be difficult for some mechanisms.
1. The Jacobian for parallel robots may be easier to find, as is the case for the Picker robot.
1. The Jacobian in robotics is particularly important because it represents the rate mapping between joint and Cartesian space. Unfortunately, for serial robots the desire is to determine the joint rates to get desired motion in Cartesian space as the tool is moved along some line in space. This usually requires the inverse of the Jacobian, which may be difficult for some mechanisms.
1. The Jacobian for parallel robots may be easier to find, as is the case for the Picker robot.