Metodología de costos aplicada a la obra pública

The gain scheduling method proposed in the previous section is simulated on a single tendon. The model uses the friction and ripple models developed in the Chapter 3. The stiness characteristics of a calibrated tendon are used to provide a realistic force/stiness displacement curve. Noise of an amplitude similar to the one observed on the real system is added through a sensor model (quantization and white noise). The test pattern consists of a force step from 10N (resp. 20N, 30N and 40N) to a force of 20N (resp. 30N,

40N and 50N) and is repeated several times. The test pattern is used in four

dierent cases:

1. Simulation with xed gains (cf. 10.3a). 2. Simulation with scheduled gains (cf. 10.3b). 3. Experiment with xed gains (cf. 10.4a). 4. Experiment with scheduled gains (cf. 10.4b).

The simulations and the experiments both conrm that the method is successful. The transient behavior of the force, that was underdamped or overdamped under the xed gain controller, is always well damped under the scheduled gain controller. Although only approximative (the partial derivative of the gains modies the pole locations), the method is intuitive and relatively easy to implement. A more detailed experimental work which

Time [s] 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 25 30 35 40 45 50 Force [N]

(a) Simulation: Fixed gains

Force [N] 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 25 30 35 40 45 50 Time [s]

(b) Simulation: Adaptive gains

Figure 10.3: Simulations: Tendon force control with/without adaptive gains. In both gures, the measured and desired tendon force is depicted. A step of 5N is commanded from dierent initial states. The adaptive controller is superior to the xed gain controller except for the lowest force which is due to the saturation of the control input.

Time [s] Force [N] 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 35 40 45 50

(a) Experiments: Fixed gains

Time [s] Force [N] 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 35 40 45

(b) Experiments: Adaptive gains

Figure 10.4: Experiments: Tendon force control with/without adaptive gains. In both gures, the measured and desired tendon force is depicted. A step of 5N is commanded from dierent initial states. The adaptive controller is superior to the xed gain controller for all the cases.

is not reported here, shows that the scheduling in β can be neglected and only α has a noticeable inuence. Unsurprisingly, α is nothing else but the tendon stiness (up to a multiplicative constant) at each working point.

10.5 Discussion

In this chapter, a tendon force controller is presented. A proportional deriva- tive controller for the tendon force using xed gains is implemented and experiments have been conducted. However, since the system is nonlinear, the controller gains can only be tuned for a specic working point and the controller is underdamped or overdamped around the nominal point. The experimental results and the simulations both conrm it. The linearization of the state dynamics allows to use a gain scheduling method that adapts the gains at each working point. The use of state dependent gains enables to design the gains by identication and to set directly the poles of a linear dif- ferential equation of the motor position error. The method only requires the derivative of the stiness curve. Experiments and simulations conrm that the controller is indeed well damped for all the working points. It should be noted, however, that vibrations appear at higher stiness mostly due to the noise introduced by the high derivatives. A limitation of the gain design is the fact that it does not account for the control input magnitude (as with all linearization or pole placement methods). Therefore, the controller should be tested on the complete working range to ensure that nonlinear eects of an input saturation are not destabilizing the plant. Indeed, at low stiness, the control eort is not very eective and a large motor displacement is needed for a small force adjustment.

11 Two time scale approach

In this chapter, a joint impedance controller is designed by building a joint controller upon the tendon force controller. Two dierent designs are pre- sented: the rst one is using the singular perturbation approach and the second one is using the cascaded approach.

The singular perturbation approach relies upon the time scale dierence between the tendon force controller and the link side dynamics. This as- sumption is similar to the one made when considering the motors as torque sources while commanding currents or voltages. In the Awiwi Hand the sti- ness is modied by the internal pretension, thus modifying the time scale dierences. Moreover, the assumption is only partially valid in the case of ngers since the links have a low inertia and the motors, together with the gear boxes, have larger inertias. It can be expected, and it is experimentally veried that the validity of the singular perturbation assumption depends on the mechanical stiness. In the rst case, the outer loop is considered as constant for the inner loop. The inner loop error is neglected arguing that, because of its speed, the inner loop is stabilized before the outer loop is disturbed. Despites its limitations it remains a good technique to approach the problem thanks to its intuitive structure.

In the second case, namely the cascaded approach, the system is brought into a cascaded form, that is, a triangular system. The stability is obtained by explicitly considering the inner loop tracking error as a forcing term for the outer loop. However, the analysis is more complex than in the singular perturbation case.

This chapter applies both methods to the case of a exible joint, the dierence being essentially visible in the stability proofs. In the rst section the dynamic model is transformed into a cascaded form. Then, the tendon force controller designed in the previous chapter is augmented with some feedforward terms and their inuence is experimentally veried. Next, the equations of a joint impedance controller are established by considering that a torque source is available at the joint. The next sections are establishing stability in the case of the singular perturbation approach and the cascaded approach. Finally, experimental results are presented. They highlight that increasing the internal pretension reduces the validity the singular perturba- tion approach.

11.1 Model

Under the assumption that the tendon force controller and the link impedance controller are working in two independent frequency domains the dynamic

equations of a nger can be written as

B ¨θ =_−ETf_t(θ_{− q0}) + τm+ b(θ, ˙θ) , (11.1)

where the link position q0 is considered to be constant w. r. t. the scale of the

motor dynamics. When considering m tendons, B ∈ Rm×m _{is a diagonal}

motor inertia matrix, θ ∈ Rm _{is the vector of the motor positions, E ∈}

Rm×m is a diagonal matrix of the inverse of the pulley radius, ft ∈ Rm

is the vector of the tendon forces. The electromagnetic torque is denoted

τm∈ Rm. Following the same approach, the link side equations are modied

to integrate the fact that the tendon forces are the input variables.

M (q) ¨q + C(q, ˙q) ˙q + g(q) + b(q, ˙q) = PTf_t+ τext . (11.2)

When considering n links, M(q) ∈ Rn×n _{is the link inertia matrix, q ∈ R}n_is

the vector of the joint positions, C(q, ˙q) ˙q ∈ Rn _{is the vector of the Coriolis}

and centrifugal terms, P ∈ Rn×m _{is the coupling matrix, f}

t ∈ Rm is the

vector of the tendon forces. The external torques and the vector of joint

frictional torques are represented by τext ∈ Rn and b(q, ˙q) ∈ Rn.