Cartesian Based Control

Introduction

In all the control schemes before, the desired trajectory is given in joint space (${\theta }_d$, ${\dot{\theta }}_d$, ${\ddot{\theta }}_d$). But you often want the end effector to follow a straight line or another path in Cartesian coordinates. Control schemes that form the errors in Cartesian space are called Cartesian based control schemes. They extend Computed Torque Control.

Approach 1: trajectory conversion

Change the Cartesian path into a joint path, then use a joint space controller:

$${\theta }_d=\text{InvKin}(X_d){\dot{\theta }}_d=J^{-1}(\theta ){\dot{X}}_d{\ddot{\theta }}_d={\dot{J}}^{-1}(\theta ){\dot{X}}_d+J^{-1}(\theta ){\ddot{X}}_d$$

Approach 2: Cartesian error

Measure the error directly in Cartesian space, with forward kinematics $X=\text{Kin}(\theta )$:

$$E_X=X_d-X$$

Force and torque relationship

The work done must be the same in any set of coordinates. Let $\delta X$ be a small Cartesian displacement of the end effector and $\delta \theta$ a small joint displacement:

$$F^T\delta X={\tau }^T\delta \theta$$

Since $\delta X=J(\theta )\delta \theta$:

$$F^{T}J(\theta )\delta \theta ={\tau }^T\delta \theta \Rightarrow F^{T}J(\theta )={\tau }^T$$ $$\boxed{\tau =J^T(\theta )F}$$

Cartesian computed torque control

Start from the joint space dynamics, $M\ddot{\theta }+C\dot{\theta }+g=\tau$, and the kinematics:

$$\dot{X}=J\dot{\theta }\ddot{X}=J\ddot{\theta }+\dot{J}\dot{\theta }\Rightarrow \ddot{\theta }=J^{-1}\ddot{X}-J^{-1}\dot{J}\dot{\theta }$$

Pre multiply the dynamics by $J^{-T}$ and use $J^{-T}\tau =F$:

$$J^{-T}M{J}^{-1}\ddot{X}-J^{-T}M{J}^{-1}\dot{J}\dot{\theta }+J^{-T}C\dot{\theta }+J^{-T}g=F$$

This gives the equation of motion in Cartesian coordinates:

$$\boxed{M_X\ddot{X}+C_X+g_X=F}$$ $$M_X=J^{-T}M{J}^{-1}{C}_X=J^{-T}[C\dot{\theta }-M{J}^{-1}\dot{J}\dot{\theta }]g_X=J^{-T}g$$

Controller

Use the same partitioning in Cartesian space:

$$F=\alpha V+\beta \alpha =M_X\beta =C_X+g_X$$

This gives $\ddot{X}=V$. The servo law is:

$$V={\ddot{X}}_d+K_v\dot{E}+K_{p}E\ddot{E}+K_v\dot{E}+K_{p}E=0$$

The joint torques are:

$$\boxed{\tau =J^T(\theta )F}$$

Note

The slide writes the servo law with $K_v$ twice. The second gain is $K_p$.