- Let . There are two Pfaffian constraints, and . Determine the appropriate number of action variables and express the differential constraints in the form .
- Introduce a phase space and convert into the form .
- Introduce a phase space and convert into the form .
- Derive the configuration transition equation (13.19) for a car pulling trailers.
- Use the main idea of Section 13.2.4 to develop a smooth-steering extension of the car pulling trailers, (13.19).
- Suppose that two identical differential-drive robots are connected together at their centers with a rigid bar of length . The robots are attached at each end of the rod, and each attachment forms a revolute joint. There are four wheels to control; however, some combinations of wheel rotations cause skidding. Assuming that skidding is not allowed, develop a motion model of the form , in which and are chosen to reflect the true degrees of freedom.
- Extend the lunar lander model to a general rigid body with a thruster that does not apply forces through the center of mass.
- Develop a model for a 3D rotating rigid body fired out of a canon at a specified angle above level ground under gravity. Suppose that thrusters are placed on the body, enabling it to be controlled before it impacts the ground. Develop general phase transition equations.
- Add gravity with respect to in Example 13.12 and derive the new state transition equation using the Euler-Lagrange equation.
- Use the constrained Lagrangian to derive the equations of motion of the pendulum in Example 13.8.
- Define a phase space, and determine an equation of the form for the double pendulum shown in Figure 13.14.
- Extend Example 13.13 to obtain the dynamics of a three-link manipulator. The third link, , is attached to the other two by a revolute joint. The new parameters are , , , , and .
- Solve Example 13.14 by parameterizing the sphere with standard spherical coordinates and using the unconstrained Lagrangian. Verify that the same answer is obtained.
- Convert the equations in (13.161) into phase space form, to obtain the phase transition equation in the form . Express the right side of the equation in terms of the basic parameters, such as mass, moment of inertia, and lengths.
- Define the Hamiltonian for a free-floating 2D rigid body under
gravity and develop Hamilton's equations.

**Implementations**

- Make a 3D spacecraft (rigid-body) simulator that allows any number of binary thrusters to be placed in any position and orientation.
- Make a simulator for the two-link manipulator in Example 13.13.

Steven M LaValle 2012-04-20