Pfaffian Constraints

A Pfaffian constraint is a set of $k$ ($k \leq n$) linearly independent constraints so that:

Where:

• $A(q) \in \mathbb{R}^{k \times n}$
• $\dot{q}$: derivative of $q$ with respect to time
• $k$: number of constraints
• $n$: number of variables needed to define robots configuration (implicit representation)

Holonomic Constraints

• A holonomic mechanical system can move in arbitrary directions (in its C-space)
• Holonomic constraints can be defined independent of $\dot{q}$ (i.e. $f(q,t)= 0$)
• Holonomic constraints reduce the dimension of the C-space (geometric limitation)
• C-space can be viewed as a surface of dimension $n-k$ embedded in $\mathbb{R}^n$
  • $n$: number of variables to define robots configuration
  • $k$: independent holonomic constraints
  • $n-k$: dimension of C-space, degree of freedom
• Integrable constraints:
  • Kinematic constraints may be integrable. In this case, the constraints are geometric constraints
  • the velocity constraints that they imply can be integrated to give equivalent configuration (geometric) constraints

Holonomic constraints $g(q(t)) = 0$ can be differentiated with respect to $t$ to yield:

Nonholonomic Constraints

• A nonholonomic mechanical system cannot move in arbitrary directions (in its C-space)
• Constraints can not be integrated
• They reduce the dimension of the feasible velocities of the system (kinematic/velocity limitation)
• They do not reduce the dimension of the reachable C-space
• e.g. rolling without slipping, differential drive, car (Ackermann steering)