This paper presents a numerical method of the minimum-time trajectory planning for a robot manipulator amid obstacles. Each joint displacement is represented by the linear combination of the finite-term quintic B-splines which are the known functions of the path parameter. The time is represented by the linear function of the same path parameter. Since the geometric path is not fixed and the time is linear to the path parameter, the coefficients of the splines and the time-scale factor span a finite-dimensional vector space, a point in which uniquely represents the manipulator motion. The displacement, the velocity and the acceleration conditions at the starting and the goal positions are transformed into the linear equality constraints on the coefficients of the splines, which reduce the dimension of the vector space. The optimization is performed in the reduced vector space using nonlinear programming. The total moving time is the main performance index which should be minimized. The constraints on the actuator forces and that of the obstacle-avoidance, together with sufficiently large weighting coefficients, are included in the augmented performance index. In the numerical implementation, the minimum-time motion is obtained for a planar 3-1ink manipulator amid several rectangular obstacles without simplifying any dynamic or geometric models.