A Vector Algebra Formulation of Mobile Robot Velocity Kinematics

Typical formulations of the forward and inverse velocity kinematics of wheeled mobile robots assume flat terrain, consistent constraints, and no slip at the wheels. Such assumptions can sometimes permit the wheel constraints to be substituted into the differential equation to produce a compact, apparently unconstrained result. However, in the general case, the terrain is not flat, the wheel constraints cannot be eliminated in this way, and they are typically inconsistent if derived from sensed information. In reality, the motion of a wheeled mobile robot (WMR) is restricted to a manifold which more-or-less satisfies the wheel slip constraints while both following the terrain and responding to the inputs. To address these more realistic cases, we have developed a formulation of WMR velocity kinematics as a differential-algebraic system—a constrained differential equation of first order. This paper presents the modeling part of the formulation. The Transport Theorem is used to derive a generic 3D model of the motion at the wheels which is implied by the motion of an arbitrarily articulated body. This wheel equation is the basis for forward and inverse velocity kinematics and for the expression of explicit constraints of wheel slip and terrain following. The result is a mathematically correct method for predicting motion over non-flat terrain for arbitrary wheeled vehicles on arbitrary terrain subject to arbitrary constraints. We validate our formulation by applying it to a Mars rover prototype with a passive suspension in a context where ground truth measurement is easy to obtain. Our approach can constitute a key component of more informed state estimation, motion control, and motion planning algorithms for wheeled mobile robots.