# Robot Arm Dynamics And Control

The authors numerically investigate the dynamics and control of an electromechanical robot arm consisting of a pendulum coupled to an electrical circuit via an electromagnetic mechanism. The analysis of the dynamical behavior of the electromechanical device powered by a sinusoidal power source is carried out when the effects of the loads on the arm are neglected. It is found that the device exhibits period-n T oscillations and high amplitude oscillations when the electric current is at its smallest value. The specific case which considers the effects of the impulsive contact force caused by an external load mass pushed by the arm is also studied. It is found that the amplitude of the impulse force generates several behaviors such as jump of amplitude and distortions of the mechanical vibration and electrical signal. For more efficient functioning of the device, both piezoelectric and adaptive backstepping controls are applied on the system. It is found that the control strategies are able to mitigate the signal distortion and restore the dynamical behavior to its normal state or reduce the effects of perturbations such as a short time variation of one component or when the robot system is subject to noises.

## Robot Arm Dynamics and Control

The working state of a system with particular dynamics can be modified because of the interaction with its environment or the application of some constraints or control laws. In this line, recent years have seen the development of various control strategies applied on electrical, mechanical, electromechanical, and even biological systems: some examples are the adaptive control [13], active control [14], the classical and active-backstepping controls [15, 16], and the sliding mode control [17]. An interesting contribution dealing with chaos control of a double pendulum arm powered through an RLC circuit is reported in [11] where the authors used the state-dependent Riccati equation control and the nonlinear saturation control techniques to suppress chaos in the dynamics of the double pendulum arm. Due to its importance for engineering and robotic applications, the control of pendulum motion has been intensively studied using various approaches, including passivity-based control [18], nonlinear control [19, 20], sliding mode control [21], motion control of two pendulums [22], and bifurcation control [23].

In this work, the dynamics and the control of an electromechanical pendulum with rigid and constant length are studied. The pendulum is coupled to an electrical part through an electromagnetic link. The dynamics considers the effect of a periodic impulsive force due to the instantaneous shock between the pendulum arm and external load masses arriving periodically. This is described mathematically by a pulse-like excitation added to the initial sinusoidal electrical excitation. In terms of the load mass, the critical electrical signal amplitude leading to the displacement of the mass is evaluated. In view of optimizing the action of the pendulum arm by counterbalancing the collision effects due to the arriving loads, a pulse-like activation signal acts periodically on the pendulum arm. Finally, an adaptive backstepping method, based on the automatic variation of an intrinsic parameter, has been developed either to control the perturbations caused by the collision of the pendulum with the load masses or to counterbalance the disturbances generated by unwanted temporal variations of some parameters of the robot device.

The work is structured as follows. Section 2 describes the electromechanical pendulum robot arm and analyzes the dynamics of the arm when the action of the load is neglected. Section 3 considers the situation where the effect of the periodic actions of the load is taken into account. In order to optimize the working conditions of the robot arm, a control strategy consisting of sending pulse-like signals following the detection of the load arrival is also considered in this section. Because the device can be subject to an unknown time variation (of regular or stochastic nature) of some of its parameters, a backstepping adaptive method is used in Section 4 to reduce the effects of these perturbations on the system. Finally, Section 5 concludes the work.

Figure 1 shows the electromechanical robot arm. It is constituted of a pendulum with a rod of length and the spherical proof mass (,), coupled electromagnetically to an electrical circuit. The electrical part of the system is constituted of a coil wired around an iron core rigidly fixed on the pendulum over a length of the pendulum whose rotation axis passes at the point O. is a constant indicating the proportion of the coiled rod length inside the magnetic field.

Figures 2(a) and 2(c) show that both the responses of the mechanical part and electrical part oscillate with perfect sinusoidal shape around the equilibrium position. This is due to the linear electromechanical coupling term that causes the mechanical and the electrical response to have the same dynamics. Moreover, the amplitudes of the dynamical variables depend on the frequency of the input signal as it appears in Figures 2(b) and 2(d). Indeed, an interesting fact appears as the frequency varies. One notes that the mechanical arm amplitude at resonance (the highest amplitude) is attained at a frequency where the electrical current is almost equal to zero. This corresponds to an optimal and recommended frequency because the consumption in terms of electrical energy is very low when the mechanical action is at its maximum level.

In order to have an extended view on the dynamical states of the robot arm, a bifurcation diagram is plotted in Figure 3 taking the excitation amplitude as the control parameter. The corresponding Lyapunov exponent is evaluated by generating the perturbation to the solutions of the dynamical equations (see (6)). If , , and are the perturbations of the dimensionless electrical current, angular displacement, and angular velocity, respectively, the perturbed solutions are , , and [25]. When they are introduced into (6) and separated (the perturbation from the dynamical variables), one obtains the following equations:Then, the Lyapunov exponent can be calculated as [25] As it appears in Figure 3, whatever the value of the parameter , the system oscillates with a period equal to where is the period of the electrical source and is an integer. Indeed, the system alternates between the period-1 and period-2 oscillation from the origin close to 0 to (see Figure 3(a)). With the parameters values of Table 1, a chaotic state is not observed as the Lyapunov exponent is always negative (Figure 3(b)). Some of the results obtained here are comparable to those of [11] where the authors found chaos and used the state-dependent Ricatti equation control and the nonlinear saturation control techniques to suppress the chaotic motion. These results are also comparable to those of [12] where the authors studied the rotation of a pendulum powered by an electromechanical excitation and observed period-1 rotation, period-1 oscillation, and period-2 oscillation.

The bifurcation diagram plotted in Figure 6 shows periodic oscillations whatever the value of the parameter ξ. But this parameter causes the system to move from the period-1 oscillations to period-2 oscillations and then period-3 oscillations. For large values of , the dynamics of the system also change to small oscillations. Thus the simple sinusoidal responses (period-) of Figure 2 are transformed into period-2 and period-3 oscillations when the value of the mass to be pushed increases.

In order to reduce the level of distortion due to the collisions, one can add a periodic pulse-like electrical voltage to the sinusoidal one in the electrical part. The period of this new voltage should be equal to that of the collisions. Practically, this can be achieved by using a (piezoelectric or optical) detector/sensor which senses the arrivals of the loads of mass and commands, through a microcontroller, for instance, the generation of the pulse-like voltage. Under these conditions, the equations of the system become is the amplitude of the electrical pulse-like voltage. In the absence of the periodic signal, one just has to set . The dimensionless forms of equations (16) arewith and other coefficients are defined in (2) and (14).

Another way to mitigate or reduce the distortion created by the collisions is to use an adaptive controller. The adaptive control is interesting as it can also mitigate the actions of other types of perturbations such as the change (continuous or stochastic) of the values of some parameters of the system. Indeed, during the functioning of the system, the environmental conditions, particularly the temperature, can force some intrinsic parameters of the system to change, for a short time or permanently, in an unknown way with regular or stochastic shape. For instance, the resistance and damping coefficients can change taking the values and , where and are the bounded perturbations due to the electrical resistance and the dissipation variations.

The dynamics of the robot arm will also change consequently. In this situation, the adaptive controller is required since it adapts its own parameters to follow the unknown variations of the system parameters. Here, we develop a backstepping control strategy that will restore the system to its regular dynamics or considerably reduce the deviation from the normal working state.

To derive the backstepping controller, let us consider the reference system equation asBecause of the perturbations, the actual state of the systems can become and satisfying (19a) and (19b), respectively. The perturbation can be of additive or multiplicative nature due, for instance, to the variations of the electric voltages or a short duration change in the system parameters. The dynamical state will deviate from its normal states and . In order to force the system to return to its normal state, one should add a control function so that the equations are now 041b061a72