This study addresses the problem of vehicle lateral and yaw motion control when both equality and inequality (i.e., bilateral and unilateral) constraints are involved. By using the Udwadia–Kalaba approach, the explicit equation of vehicle motion with equality constraints is established, and the corresponding control inputs can be obtained from the equation. The equality constraints aim to render the vehicle to move along the desired trajectory. However, as the initial conditions of vehicle motion may take values leading the vehicle to violate the road-bound lines, it is necessary to impose an additional constraint to constrain the vehicle to move within the road-bound lines, which is an inequality constraint. As the inequality constraint cannot be handled by the original Udwadia–Kalaba approach, a creative diffeomorphism approach is proposed to integrate the inequality constraint into the equality constraints, and thus it creatively enables the Udwadia–Kalaba approach to deal with both equality and inequality constraints. By solving the equation established based on the Udwadia–Kalaba approach and diffeomorphism approach, the control inputs that can render the vehicle to move along the desired trajectory without violating the road-bound lines are obtained. The effectiveness of the proposed method is demonstrated by numerical simulation results.