Inverse Kinematics

Textbook Chapter 6

Robot convenience wrappers

For simplified calls using a Robot model (e.g., inverse_kinematics_body(robot, target, guess, ε_ω, ε_v)), see Robot Model.

ModernRoboticsBook.inverse_kinematics_body — Method
inverse_kinematics_body(body_screw_axes, home_ee_pose, target_config, initial_guess, angular_tolerance, linear_tolerance)

Computes inverse kinematics in the body frame for an open chain robot using Newton-Raphson iteration.

Why is inverse kinematics iterative?

Unlike forward kinematics (which has a closed-form solution), inverse kinematics — finding joint angles for a desired end-effector pose — generally has no closed-form solution for arbitrary robots. This function uses Newton-Raphson iteration, repeatedly computing the error and updating the joint angles via the Jacobian until convergence.

Arguments

  • body_screw_axes: the joint screw axes $B_i$ expressed in the end-effector (body) frame at the home configuration, as a $6 \times n$ matrix.
  • home_ee_pose: the $4 \times 4$ end-effector pose $M \in$ SE(3) when all joint positions are zero (home configuration).
  • target_config: the desired $4 \times 4$ end-effector pose $T \in$ SE(3).
  • initial_guess: an $n$-vector initial guess of joint positions $\theta_0$.
  • angular_tolerance: small positive tolerance on the end-effector orientation error.
  • linear_tolerance: small positive tolerance on the end-effector position error.

Returns

A tuple (joint_positions, success) where joint_positions is the $n$-vector solution and success is a Bool.

Examples

julia> body_screw_axes = [  0  0 -1  2  0  0   ;
                            0  0  0  0  1  0   ;
                            0  0  1  0  0  0.1 ]';

julia> home_ee_pose = [ -1  0  0  0 ;
                         0  1  0  6 ;
                         0  0 -1  2 ;
                         0  0  0  1 ];

julia> target_config = [ 0  1  0     -5 ;
                         1  0  0      4 ;
                         0  0 -1 1.6858 ;
                         0  0  0      1 ];

julia> initial_guess = [1.5, 2.5, 3];

julia> angular_tolerance, linear_tolerance = 0.01, 0.001;

julia> inverse_kinematics_body(body_screw_axes, home_ee_pose, target_config, initial_guess, angular_tolerance, linear_tolerance)
([1.5707381937148923, 2.999666997382943, 3.141539129217613], true)
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ModernRoboticsBook.inverse_kinematics_space — Method
inverse_kinematics_space(screw_axes, home_ee_pose, target_config, initial_guess, angular_tolerance, linear_tolerance)

Computes inverse kinematics in the space frame for an open chain robot using Newton-Raphson iteration.

Body vs space IK

This is the space-frame version of inverse_kinematics_body. Both converge to the same solution; the difference is whether the error twist is computed in the body or space frame. The initial guess matters — the algorithm may converge to different solutions or fail to converge depending on the starting point.

Arguments

  • screw_axes: the joint screw axes $S_i$ expressed in the space (base) frame at the home configuration, as a $6 \times n$ matrix.
  • home_ee_pose: the $4 \times 4$ end-effector pose $M \in$ SE(3) when all joint positions are zero (home configuration).
  • target_config: the desired $4 \times 4$ end-effector pose $T \in$ SE(3).
  • initial_guess: an $n$-vector initial guess of joint positions $\theta_0$.
  • angular_tolerance: small positive tolerance on the end-effector orientation error.
  • linear_tolerance: small positive tolerance on the end-effector position error.

Returns

A tuple (joint_positions, success) where joint_positions is the $n$-vector solution and success is a Bool.

Examples

julia> screw_axes = [  0  0  1  4  0  0   ;
                       0  0  0  0  1  0   ;
                       0  0 -1 -6  0 -0.1 ]';

julia> home_ee_pose = [ -1  0  0  0 ;
                         0  1  0  6 ;
                         0  0 -1  2 ;
                         0  0  0  1 ];

julia> target_config = [ 0  1  0     -5 ;
                         1  0  0      4 ;
                         0  0 -1 1.6858 ;
                         0  0  0      1 ];

julia> initial_guess = [1.5, 2.5, 3];

julia> angular_tolerance, linear_tolerance = 0.01, 0.001;

julia> inverse_kinematics_space(screw_axes, home_ee_pose, target_config, initial_guess, angular_tolerance, linear_tolerance)
([1.57073782965672, 2.999663844672525, 3.141534199856583], true)
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