How do I find the singular configurations of the Stewart platform using SimMechanics?

I would like to be able to determine apriori the singular configurations of my Stewart platform. Since I am designing a controller for controlling the plate motion, it becomes important that the controller be aware of these singularities. The "robust singularity handling" option is very helpful in the simulation, but I have no way of incorporating this in the actual implementation of my controller. It becomes imperative that I have access to the singular configurations apriori.

Answer 1

[MathWorks Support Team]

There is no way to access the singular configurations for a machine in SimMechanics.

1. The solution that follows here assumes that the machine in question is a manipulator, e.g. a Stewart platform. For a manipulator, singularities pose certain problems. "End-effector" will refer to the plant-frame. For the Stewart platform, it refers to the plate, since the plate is controlled by the actuators.

• If you would like to calculate the position and velocity of the end-effector of the manipulator when torques and forces are applied at the joints/end-effector, then the problem requires forward dynamics (Forward Dynamics mode in SimMechanics).
• If you would like to calculate the actuator torques and forces that should be applied to the manipulator to produce a given velocity profile of the end-effector, then the problem requires inverse dynamics (in SimMechanics, Inverse Dynamics mode for open-topology machines, Kinematics mode for closed-loop machines).

2. There are two kinds of singularities that you can encounter in a manipulator - kinematic and actuator.

a. Kinematic singularities will not allow you to back-calculate the joint velocities from the end-effector velocities.

Kinematic singularities refer to those configurations of the manipulator in which a motion of the end-effector cannot be accommodated by motions of the joints. Please note that the Stewart platform does not have kinematic singularities. Any motion of the end-plate can be accommodated by a suitable change of the joint coordinates.

b. Actuator singularities will not allow you to control the motion with the joint actuators of the end-effector along some directions.

An actuator singularity occurs when the actuators of the manipulator cannot produce end-effector forces and torques in some directions. Thus, the manipulator will not be able to control motion along those directions. Such a configuration occurs in the Stewart platform when the plant-frame (i.e., the plate) is coplanar with the base-frame (the plane containing the base). In this case, the actuators cannot produce or resist forces which are non-coplanar.

If the Stewart platform were to encounter an actuator singularity, then the controller with the actuators will not be able to control the motion of the platform along certain directions. Your controller would fail no matter how good your design was.

To calculate the actuator singularities, try the following:

The system of wrenches that can be applied to the plant-frame are zero-pitch wrenches because the actuators are force-actuators. In any configuration of the Stewart platform, these wrenches/screws coordinates can be calculated. For an N-legged Stewart platform, one would need to form a 6xN manipulator Jacobian. The rank of the manipulator Jacobian can be used as a check to determine the singular configuration. For example, if the goal is to produce an arbitrary wrench (i.e three forces and three torques) at the plate, the rank of this Jacobian should be 6.