3.4 Inverse Differential Kinematics

Can we just invert $J$? not unless $n=m$, the DoF in joint and task space respectively with full rank. You can take a pseudo-inverse if that isn’t the case but it isn’t perfect and may not be numerically stable.

When this happens it is called a singularity. Singularities should be avoided as the solution “explodes” and joint speeds go to infinity and robots can become unsafe. In order to work around this we can add damping and minimize error.

$||J\Delta q-e||+{\lambda }^2||\Delta q|{|}^2$ where $\lambda$ is constant. This is known as the Damped Least-Squares method.

3.4.1 Inverse Kinematics of Mobile Robots.

There is no unique relationship between the amount of rotation of a robots wheels and its position in space for non-holonomic robots. Instead, this is a problem about solving for the velocities within the local robot coordinate frame.

If we take our expression from the last section, ${\dot{\xi }}_I=\text{prescript}IRT(\theta ){\dot{\xi }}_R$ and multiply both sides by $T^{-1}(\theta )$ we can find a relationship between the angular speeds for each wheel and a desired ${\dot{x}}_R$ and ${\dot{\theta }}_R$.

Note to self; there is more in this section about generalizing to omni-directional wheeled robots. good fucking lord I’m not copying that over right now, please just go look at the textbook. thanks

3.4.2 Feedback control for mobile robots

Assume you have a position for a robot $x_r$, $y_r$, ${\theta }_r$ and a goal position $x_g$, $y_g$, ${\theta }_g$. We can calculate the error in the desired pose

$\rho =\sqrt{(x_r-x_g{)}^2+(y_r-y_g{)}^2}$ $\alpha =ta{n}^{-1}\frac{y_g-y_r}{x_g-x_r}-{\theta }_r$ $\eta ={\theta }_g-{\theta }_r$ using a simple proportional controller with gains $p_{1,}{p}_{2,}{p}_3$ $\dot{x}=p_1\rho$ $\dot{\theta }=p_{2\alpha }+p_3\eta$ which will let the robot drive in a curve until it reaches the desired pose.

3.4.3 Under-actuation and Over-actuation

given: $n$ - the number of DoF in joint space $m$ - the number of DoF in task space

• $n<m$ - The Robot is under-actuated and the problem is kinematically deficient. The Jacobian $J$ is “wide” and the only way to solve the inverse kinematics problem is through the pseudo inverse $J^{+}$
• $n=m$ - The Robot is fully actuated. The Jacobian $J$ is square and full rank meaning that the forward kinematics equation is directly invertible. The world is great!
• $n>m$ - The Robot is over-actuated and the problem is kinematically redundant. the only way to solve the problem is through the psuedo-inverse. The redundancy is a benefit though - it means there are more solutions for each desired pose. The human arm, for example, is over-actuated with 7 DoF (3 in the shoulder and wrist, 1 in the elbow) in a 6 DoF task space (3d real space)