A general solution based on time-variant NURBS is presented for robot position and orientation planning.Adding an extra component to the geometric NURBS, the time -variant NURBS model for motion design is set up. It preserves most properties, algorithms and tools in NURBS theory. Proper motion curve degrees are determined. Base on the motion model, an interpolation method that can consider constraints of different orders is presented. It allows the designer to assign the key point any one or more constraints such as position, velocity and acceleration, etc. The motion model and interpolation method are applicable to 1D and 2D motion design with dimensional reduction of control point matrix. A welding torch motion planning is given as an example.