Kinematics

Forward kinematics

Forward kinematics (FK) is the pose of the end-effector given the joint coordinates. It can be computed for robots of the DHRobot or ERobot class

T = robot.fkine(q)

where T is an SE3 instance.

T = robot.fkine_all(q)

where T is an SE3 instance with multiple values, the pose of each link frame, from the base T[0] to the end-effector T[-1].

Inverse kinematics

Inverse kinematics (IK) is the joint coordinates required to achieve a given end-effector pose. The function is not unique, and there may be no solution.

Method	MATLAB version	Type	Joint limits	Description
ikine_a	ikine6s	analytic	no	For specific DHRobots only
ikine_LM	ikine	numeric	no	Levenberg-Marquadt with global search option
ikine_LMS		numeric	no	Levenberg-Marquadt with Sugihara's tweak
ikine_min	ikunc	numeric	no	Uses SciPy.minimize, user cost function, stiffness
ikine_min(qlim=True)	ikcon	numeric	yes	Uses SciPy.minimize, user cost function, stiffness

All methods take optional method-specific arguments and return a named tuple

sol = ikine_XX(T)

The elements of the tuple sol include at least:

Element	Type	Description
q	ndarray (n)	Joint coordinates for the solution, or None
success	bool	True if a solution found
reason	str	reason for failure
iterations	int	number of iterations
residual	float	final value of cost function

Analytical solutions

Numerical solutions

Performance

6DOF robot

puma = rtb.models.DH.Puma560()
T = puma.fkine(puma.qn)
sol = puma.ikine_XX(T)
print("residual = ", np.linalg.norm(T - puma.fkine(sol.q)))

Method	Time (ms)	Residual
ikine_a	0.34	1.5e-16
ikine_LM	6.9	8.8e-11
ikine_LMS	3.8	2.2e-6
ikine_unc	45	1.3e-7
ikine_min	1500	1.0e-5
ikine_min(qlim=True)	131	1.0e-5

7DOF robot

puma = Panda()
T = SE3(0.7, 0.2, 0.1) * SE3.OA([0, 1, 0], [0, 0, -1])
sol = puma.ikine_XX(T)
print("residual = ", np.linalg.norm(T - puma.fkine(sol.q)))

Method	Time (ms)	Residual
ikine_LM	9.1	1.5e-11
ikine_LMS	11	3.6e-6
ikine_min	75	4.5e-8
ikine_min(qlim=True)	1600	1.6e-7

The SciPy based mimimizers are slow because they use a scalar cost measure and must numerically compute a Jacobian in order to solve.

Robots with multiple end effectors

This is just speculation so far...

T = robot.fkine(q, link)

where link is either a reference to a Link subclass object, or the name of the link.

q = robot.ikine_XX(T, link)

where link is either a reference to a Link subclass object, or the name of the link.

Ideally we would be able to express other constraints by passing in a callable

q = robot.ikine_XX(T, lambda q: 0.1 * q[2]**2)

which adds a cost for elbow bend, for example.