Velocity Kinematics and Statics

Textbook Chapter 5

Robot convenience wrappers

For simplified calls using a Robot model (e.g., jacobian_space(robot, q)), see Robot Model.

ModernRoboticsBook.jacobian_body — Method
jacobian_body(body_screw_axes, joint_positions)

Computes the body Jacobian $J_b(\theta)$ for an open chain robot.

What is the Jacobian?

The Jacobian relates joint velocities to end-effector velocity: $V_b = J_b(\theta) \dot{\theta}$. It tells you how fast and in which direction the end-effector moves for a given set of joint velocities. It is also used to map wrenches (forces/torques) from the end-effector back to the joints.

Arguments

  • body_screw_axes: the joint screw axes $B_i$ in the end-effector (body) frame at the home position, as a $6 \times n$ matrix.
  • joint_positions: an $n$-vector of joint positions $\theta$.

Returns

The $6 \times n$ body Jacobian $J_b(\theta)$.

Examples

julia> body_screw_axes = [0 0 1   0 0.2 0.2;
                          1 0 0   2   0   3;
                          0 1 0   0   2   1;
                          1 0 0 0.2 0.3 0.4]';

julia> joint_positions = [0.2, 1.1, 0.1, 1.2];

julia> jacobian_body(body_screw_axes, joint_positions)
6×4 Matrix{Float64}:
 -0.0452841  0.995004    0.0       1.0
  0.743593   0.0930486   0.362358  0.0
 -0.667097   0.0361754  -0.932039  0.0
  2.32586    1.66809     0.564108  0.2
 -1.44321    2.94561     1.43307   0.3
 -2.0664     1.82882    -1.58869   0.4
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ModernRoboticsBook.jacobian_space — Method
jacobian_space(screw_axes, joint_positions)

Computes the space Jacobian $J_s(\theta)$ for an open chain robot.

Body vs space Jacobian

The space Jacobian $J_s$ gives the end-effector velocity in the fixed space frame, while the body Jacobian $J_b$ gives it in the end-effector's own frame. They are related by $J_s = [\text{Ad}_{T_{sb}}] J_b$. Use whichever matches the frame your task is defined in.

Arguments

  • screw_axes: the joint screw axes $S_i$ in the space frame at the home position, as a $6 \times n$ matrix.
  • joint_positions: an $n$-vector of joint positions $\theta$.

Returns

The $6 \times n$ space Jacobian $J_s(\theta)$.

Examples

julia> screw_axes = [0 0 1   0 0.2 0.2;
                     1 0 0   2   0   3;
                     0 1 0   0   2   1;
                     1 0 0 0.2 0.3 0.4]';

julia> joint_positions = [0.2, 1.1, 0.1, 1.2];

julia> jacobian_space(screw_axes, joint_positions)
6×4 Matrix{Float64}:
 0.0  0.980067  -0.0901156   0.957494 
 0.0  0.198669   0.444554    0.284876 
 1.0  0.0        0.891207   -0.0452841
 0.0  1.95219   -2.21635    -0.511615 
 0.2  0.436541  -2.43713     2.77536  
 0.2  2.96027    3.23573     2.22512
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