Velocity versus torque control

As mentioned before, Archie is equipped with a combination of brushed (permanent-magnet) DC and brushless DC motors. These motors can be described with a same mathematical model. The armature (rotor) current can be described as the following differential equation:

𝐿^𝑑𝐼_𝑑𝑡^𝑎+𝑅𝐼_𝑎 = 𝑉_𝑎− 𝑉_𝑔 5-9 where 𝐿 is the armature inductance, 𝑅 is the armature resistance, 𝑉𝑎 and 𝐼𝑎 are voltage and current of the armature, respectively and 𝑉𝑔 is the back emf (electromotive force) voltage generated in the armature of the motor which is proportional to the angular speed of the motor 𝜔𝑚 :

𝑉_𝑔 = 𝑘_𝑣𝜔_𝑚 5-10 where 𝑘𝑣 is the voltage constant that depends on the characteristic of the motor and the magnetic flux of the motor coil.

The torque produced by the motor 𝜏_𝑚 is also proportional to the armature current 𝐼_𝑎 through torque constant 𝑘𝑡 :

𝜏𝑚 = 𝑘𝑡𝐼𝑎 5-11 It can be noted that in the SI unit, the numerical values of 𝑘_𝑡 and 𝑘_𝑣 are the same (de silva, 2007).

Fig. 5.4 shows a set of torque-speed curves versus various applied voltage that is useful to determine the torque constant. The rated voltage 𝑉𝑟 is corresponding to the blocked rotor (stall) torque 𝜏₀ when motor is stalled.

The mechanical equation of motion of the rotor can be described as:

𝜏_𝑚 =𝐼_𝑚^𝑑𝜔_𝑑𝑡^𝑚+𝐹_𝑚𝜔_𝑚+𝜏_𝑙 5-12 where 𝜏𝑙 is the reaction torque exerted from manipulator, 𝐼𝑚 and 𝐹𝑚 represent the moment of inertia and the viscous friction coefficient at the motor shaft, respectively. By substituting and transforming to the Laplace domain we have the following motor equation:

𝑘𝑡𝐼𝑎 = (𝑠𝐼𝑚+𝐹𝑚)𝜔_𝑚+𝜏𝑙 5-13 𝑉𝑎 = (𝑠𝐿+𝑅)𝐼_𝑎+𝑘𝑣𝜔𝑚 5-14

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Fig. 5.4 General torque-speed curves of DC motor

Fig. 5.5 shows the block diagram of the above mentioned equation of DC motor in Laplace domain. Considering that the value of the electric time constant 𝐿 𝑅⁄ is very small in comparison with mechanical time constant 𝐼𝑚⁄𝐹𝑚 leads to neglect the electric time constant in motor model. This is common assumption that is correct for the most of DC motors.

Fig. 5.5 Block diagram of DC motor

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Now , the reduced order model of motor can be described in matrix form for whole manipulator by:

𝑲_𝑟⁻¹𝝉=𝑲_𝑡𝒊_𝑎 5-15 𝒗_𝑎 =𝑹𝒊_𝑎 +𝑲_𝑣𝒒̇_𝑚 5-16 where 𝒊_𝑎 is a vector of the armature current, 𝑲_𝑡 is the diagonal matrix whose elements are torque constants, 𝒗𝑎 is a vector of the armature voltage, 𝑲𝑣 is the diagonal matrix of the voltage constants, 𝑹 is the diagonal matrix of the armature resistances of the 𝑛 motors.

Furthermore, any electric motor needs power amplifier (electric drive) to generate required voltage for the motor by amplifying the power source voltage. Then the relation between the vector of armature voltage 𝒗𝑎 and the vector of control voltage 𝒗𝑐 of the 𝑛 servomotor can be written as

𝒗_𝑎 =𝑮_𝒗𝒗_𝑐 5-17 where 𝑮𝒗 is the diagonal matrix of gains of the 𝑛 amplifiers. Substituting the last three equations on each other leads to find the vector of joint torques as:

𝝉=𝑲_𝑟𝑲_𝑡𝑹⁻¹(𝑮_𝒗𝒗_𝑐 − 𝑲_𝑣𝑲_𝑟𝒒̇) 5-18 The block diagram of the system are depicted in Fig. 5.6.

Fig. 5.6 Block diagram of the manipulator and motor system as voltage-controlled system(Siciliano et al., 2009)

Now, one can specify the required control voltage corresponding to 𝒗𝑐 corresponding to the desired joint velocity vector 𝒒̇_𝑑 and desired vector of joint torques 𝝉 based on the following relation:

𝒗𝑐 =𝑮𝒗−𝟏𝑲𝑟−𝟏𝑲𝑡−𝟏𝑹 𝝉𝒅+𝑮𝒗−𝟏𝑲𝑣𝑲𝑟𝒒̇𝑑 5-19

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To control the above system two cases can be taken into account:

1. The first case is when the gear ratio of drives, i.e. diagonal elements of the matrix 𝑲𝑟 are large and the values of motor resistance, i.e. elements of the matrix 𝑹 are very small and assuming the values of the joint torques required for performing the desired motion are not very large. In this case the first term of the control voltage 𝑮_𝒗^−𝟏𝑲_𝑟^−𝟏𝑲_𝑡^−𝟏𝑹 𝝉_𝒅 is negligible.

Then control voltage is reduced to

𝒗_𝑐 ≈ 𝑮_𝒗^−𝟏𝑲_𝑣𝑲_𝑟𝒒̇_𝑑 5-20 This scheme is called velocity or voltage control method since the control voltage is achieved according to the desired velocity. It is noted that the relationship between voltage 𝒗𝑐 and speed 𝒒̇_𝑑 is independent of manipulator parameters. Therefore the control system is robust with respect to the parameter variation of manipulator model. The greater the value of the gear ratios, the more robust to the parameter variation.

Furthermore, due to the fact that the matrix 𝑮_𝒗^−𝟏𝑲_𝑣𝑲_𝑟 is diagonal, the control voltage of each joint only depends on the speed of the same joint and not depends on the speed of the other joints. Hence a decentralized control method can be used for position control of the joint in this case, since each joint can be controlled independently.

2. In the second case, on the other hand, the required joint torques for desired motion control are large or the system is direct-drive (𝑲_𝑟 = 𝑰 ). In this case, the first term of control voltage is no longer negligible. Therefore it is needed to specify the required joint torques 𝝉_𝒅(𝑡) to track any desired motion in terms of the joint accelerations 𝒒̈(𝑡) , velocities 𝒒̇(𝑡) and positions 𝒒(𝑡). This can be done using inverse dynamics technique that requires the accurate knowledge of the manipulator dynamic model. In this manner, the control system is needed to be a centralized control method, because computing the torque history at each joint requires to know the time evolution of the motion of all the joints.

According to the above mentioned relation, the control voltage in this case is determined based on the desired torque values and desired joint velocities:

𝒗_𝑐 =𝑮_𝒗^−𝟏𝑲_𝑟^−𝟏𝑲_𝑡^−𝟏𝑹 𝝉_𝒅+𝑮_𝒗^−𝟏𝑲_𝑣𝑲_𝑟𝒒̇_𝑑 5-21 Since the matrices 𝑲_𝑡^−𝟏, 𝑲_𝑣 and 𝑹 are related to the characteristic of the motor and changing according to the different operation conditions of the motors, the motor control can be described as a current control instead of voltage control. In this case, the control system is less sensitive to the parameter variations of the motors. The equation of the actuator that perform as a torque-controlled generator can be specified as:

𝒊𝑎 = 𝑮𝒊𝒗𝑐 5-22 where 𝑮𝒊 is the constant diagonal matrix relates the armature currents 𝒊𝑎 and the control voltages 𝒗_𝑐. Therefore the joint torques are derived as

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𝝉 =𝑲_𝑟𝑲_𝑡𝑮_𝒊𝒗_𝑐 5-23 As a consequence, the vector of the voltage control for a torque-controlled method is obtained by

𝒗_𝑐 = 𝑮_𝒊^−𝟏𝑲_𝑟^−𝟏𝑲_𝑡^−𝟏 𝝉_𝒅 5-24 where desired joint torques 𝝉𝒅 are obtained by computation of the inverse dynamics for a desired motion. Although, the centralized control method seems to be a feedforward system, the use of error between the actual and desired trajectory is necessary. Because the dynamic model of the system, even though a very complicated one, is anyhow idealization of reality which does not include dimension tolerances, friction and gear backlash and also uses simplified assumption such as link rigidity.

With comparing two cases, The first case assumptions are more compatible with Archie characteristic, since the values of gear ratios used in Archie are large, i.e. 160 for brushless and 450 for brushed DC motor, and the resistance of utilized DC motors are very small and also the required joint torques needed for walking are not very large. Thus velocity control (voltage control) method is used in each joint of Archie that is compatible with decentralized control strategy. So the next section explains the independent joint control strategies used for decentralized control.