Robots are complex electromechanical systems where several electric drives are used to control the movement of articulated structures.
This case study presents the modeling and simulation of a six-degrees-of-freedom robot manipulator. The two main joints models are built using brushless DC motor drives that are connected to the rest of the manipulator through speed reducers. The control system, which consists essentially of two position control loops, is built with Simulink® blocks. The inner speed and torque control loops are already included in the drive model. The rest of the manipulator and its load are represented by two Simulink nonlinear models, one for each motor drive.
The operation of the joints using typical trajectories is simulated and results are presented.
The robot considered in this example is a general-purpose six-degrees-of-freedom robot manipulator (GMF S-360) of parallelogram linkage type. Six-Degrees-of-Freedom Robot Manipulator shows the structure of the robot and its workspace. The robot has six axes. The three axes (Θ1, Θ2, Θ3) shown in the figure are for arm positioning and the others (α, β, γ) are for orientation of the end effector. In the horizontal plane, the robot can cover a 300 degree arc (Θ1 = -150° to Θ1 = 150°).
The robot's axes are driven by brushless DC motors that are modeled by permanent-magnet synchronous motors fed by PWM inverters (AC6 drive model). Speed reducers of belt type and gearbox are used to transmit torque from the motors to the joints.
Six-Degrees-of-Freedom Robot Manipulator
We will consider in particular the two first joints (axis 1 and axis 2), which drive the entire robot and its load. The first axis uses a 2 kW brushless DC motor and a 1:130 speed reducer. The second axis uses a 1 kW brushless DC motor and a 1:100 speed reducer. Brushless DC Motor Drive for Position Control of Robot Joint shows a simplified diagram of the position control system for one robot link.
The control system consists of three control loops connected in a cascade configuration: an outer position loop includes an inner speed control loop and an innermost current control loop. The PM synchronous motor is fed by a three-phase PWM inverter operating in current-controlled mode. Field-orientation scheme is used to decouple the variables so that flux and torque can be separately controlled by stator direct-axis current ids and quadrature-axis current iqs, respectively. The quadrature-axis current reference iqs* (which represents the torque command) is provided by the speed control loop. The direct-axis current reference ids* is kept equal to 0.
A speed/position sensor is used to provide the information required by the speed and position control loops. The rotor position is also required for coordinates conversion (dq to abc).
Each motor drives the rest of the robot structure, including the other links and the load, through a speed reducer.
Brushless DC Motor Drive for Position Control of Robot Joint
The entire drive system for the robot's two first joints, including motor drives, speed reducers, equivalent loads, and controllers is available in the Robot Axis Control Using Brushless DC Motor Drives example.
The brushless DC motor drives are represented by two AC6 (PM Synchronous Motor Drive) blocks. This block models a complete brushless DC motor drive including a permanent-magnet synchronous motor (PMSM), an IGBT inverter, speed controller, and current controller. The AC6 inputs are the speed commands and the outputs are the motor speed, which are fed to the inputs of the speed reducers.
The speed reducers are modeled by two Speed Reducer blocks. The inputs for these blocks are the motors' speeds, and the outputs are the torques from the low-speed sides, which are applied to the robot structure model. The speed reducers are characterized by their ratio and inertia and the stiffness and damping of input and output shafts.
The speed reducers' output shafts are connected to the T1 and T2 inputs of a Robot block that represents the rest of the robot structure. This block calculates the effective torque reflected to each joint. For each joint (numbered i), we can consider globally the other links effects as a single load reflecting to the joint a torque that is composed of three terms
where Θi is joint angular position, Ji is inertia, Ci is centrifugal and Coriolis coefficient, and Gi is gravitational coefficient.
The Robot model is built with Simulink blocks.
In this diagram, the parameters J1, C1, G1, J2, C2, and G2 are functions of joint positions. Implement them by using polynomials or lookup tables.
The joint positions Θ1 and Θ2 are controlled by outer control loops that force Θ1 and Θ2 to follow the references imposed by the trajectories of the manipulator. Various algorithms can be used for these control loops. The most popular ones are proportional-derivative, computed torque, and adaptive. In this example, proportional-derivative controllers are implemented for both axes.
Cubic polynomial test trajectories for robot motion are generated by the Trajectory Generator block.
The test trajectories consist of a movement from position 6 to position 3 in the workspace (Θ2 varying from -π/4 to π/4) while rotating around axis 1 from one position to another (Θ1 varying from -π/6 to π/6). The parameters to be specified for this block are initial position [Θ1ini, Θ2ini], final position [Θ1fin, Θ2fin], and move time. The following figure shows the changes of robot structure during the programmed movement.
The variation of inertia due to structure changes is reflected to axis 1 as an inertia varying as a function of Θ2 (from 215 kgm2 to 340 kgm2 passing by a minimum of 170 kgm2). The inertia reflected to axis 2 is a constant (J2 = 50 kgm2). These inertia variations are represented by nonlinear functions implemented in the Robot block.
The test trajectories described above constitute one of the most demanding trajectories for the motor drive of the first and second joints. They are used here to evaluate the tracking performance of the two electric drive systems.
In the example, the manipulator is programmed to rotate from -30° to 30° during 1.5 seconds, and at the same time the arm is moved from the back position (Θ2 = -45°) to the most advanced position (Θ2 = 45°). The simulation is run using a time step of 1 µs.
The responses of the manipulator and the motor drives 1 and 2 are displayed on three scopes connected to the output variables of the AC6 and Robot blocks.
During the movement, the joint positions Θ1 and Θ2 follow the imposed cubic trajectories with low tracking error.
The brushless DC motor drives behave very well during the test trajectories. The DC bus voltages are maintained at relatively constant levels during the deceleration of the motors. The developed torques are proportional to the motor currents' amplitudes. This demonstrates the good operation of the field-oriented control algorithms.
 Miller, T. J. E., Brushless Permanent-Magnet and Reluctance Motor Drives, Clarendon Press, Oxford, 1989.
 Spong, M. W., and Vidyasagar, M., Robot Dynamics and Control, John Wiley & Sons, New York, 1989.