# In path planning, what kind of path is feasible for a nonholonomic robot?

For a car-like nonholonomic robot, What kind of path is feasible?

Is it right that a path is feasible for a car-like robot if it's curvature continuous, and the curvature on any point doesn't exceed the turning limit of the robot?

I'm not quite sure about this because I noticed some paper said their path planning algorithms deal with the differential constraint of the vehicle. I don't understand what this means. I think if a path is smooth enough, the vehicle should be able to track this path exactly (ignoring the dynamic model).

So what kind of path satisfies the differential constraints of vehicles? Or what kind of path doesn't satisfy the differential constraints of vehicles? I really need some help here. Thanks!

## 2 Answers

For a skid-steered vehicle, the constraint has this form: xdot * sin(th) - ydot * cos(th) = 0. All trajectories must satisfy this equation. The robot can in fact spin on a dime because such motion has (xdot,ydot)=0. It can also parallel-park, which is not a smooth path in the (x,y) plane.

A non-holonomic vehicle is constrained in 3 degrees of freedom (x, y, theta) where y_dot = 0. A non-holonomic vehicle can therefore only translate along a curved path governed by its kinematics (skid steer differential is different from Ackerman differential). You are correct in saying that a non-holonomic vehicle can only have a feasible path that lies on a curved path from the robots current pose.

When a paper says deal with the differential constraint of the vehicle they mean that the robot cannot, for example, rotate on the spot (skid steer) or actuate in the y or z direction. This is important in path planning because there is no point in considering paths the robot cannot achieve.

• Thank you for answering my question! It truly helps. – 李江南 Oct 6 '19 at 14:01
• The vehicle's kinematics aren't fixed but they are the result of a kinematic model. In the system identification phase, it's determined what the physical limits of the robot are. If the robot can move it's wheels separately, the allowed maneuver are endless, which means the only restriction is the imagination of the human operator how to steer the car. – Manuel Rodriguez Oct 7 '19 at 20:12
• Yeah ok, except that OP specifically asked about differential kinematics and the curved path constraint? – Grant Dare Oct 7 '19 at 21:05