9 D I S C U S S I O N
can produce a desired output given a certain input. In this context, the creation of an antagonistic joint, a virtual muscle, a force controller or a bioinspired waveform generator can be seen as means to engineer new systems which are able to produce a desired output in terms of their tracking performance, force generation or stall-force maximization.
As far as the specific contributions of this work are concerned, these are recapitulated in this section together with the main objectives of this thesis according to their order defined in sect.1.2.
Development of a motor model
The main objective of this work was to develop a model of the walking piezoelectric motor which can faithfully reproduce its dynamic behav-ior, especially under the influence of an external load. This objective was motivated by the desire to employ the motor in a force control sce-nario and the lack of any publicly available motor models of this kind.
Macroscopically, the motor exhibits several non-linear phenomena including changing motor characteristics due to the choice of driving signals, frequency-dependent stall-force limits and decreasing drive velocity under load. Since the working principle of the motor relies on discontinuous frictional interaction between piezoelectric bimorph elements (legs) and a ceramic bar (drive rod), the practical derivation of a motor model is hampered by the difficulty to obtain experimen-tal data of this interaction from a fully-assembled motor during its operation. In fact, the only measurements available to the author were the tangential position of the drive rod and the magnitude of the tangential load. Based on these measurements, two motor models have been developed within the scope of this thesis.
The first model (see chapt.3) is based on the analytic approach and describes the low-level frictional interactions between the legs and the drive rod by means of several physically meaningful assumptions with ten unknown model parameters (see sect. 3.5). The feasibility of the modeling assumptions is confirmed in a global optimization process in which the unknown model parameters are identified and result in a motor model which can fully explain the experimentally measured data. Furthermore, clusters of physically meaningful parameter values are found as a side effect of the optimization process which is a strong indicator for a meaningful choice of model parameters.
The derived model is capable of reproducing the observed non-linear phenomena in the operation of the walking motor within the full bandwidth of its rated operation. In particular, the velocity of the motor does not change proportionally to the level of deflection of the legs when different driving signals are used. This phenomenon is explained by the introduction of a hysteretic nonlinearity, which is motivated by the existence of ferroelectric hysteresis in the piezo-electric material (see sect.3.3.3). Also the frequency-dependent stall force limits and non-linear velocity decrease under load is faithfully
reproduced by the motor model. The nature of these phenomena is more complex since it involves frictional interactions in both the static and dynamic domains. For this reason, the choice of a suitable friction model is of paramount importance for the analytic motor model. Sev-eral extensions of the Coulomb friction model have been considered but could not reproduce the experimental data. These models incor-porate a discontinuity at zero velocity crossing resulting in a strong variation of the frictional forces due to the transition between the static and dynamic part of the friction model. However, this behavior is an artifact of the oversimplified models. The transition between the static and dynamic regimes has been shown to be rather displacement- than velocity-dependent [168]. Furthermore, the increased level of static friction as compared to dynamic friction depends on the contact time between the surfaces [160]. These findings speak for the inapplicability of the discontinues models to describe the frictional phenomena in the walking motor, especially under high-frequency operation. The final choice of the LuGre friction model seems to be appropriate since the LuGre model reproduces the stick-slip transitions in a continuous manner. In this work, the LuGre model was extended in order to include the impact dynamics of the legs and the changing friction levels during the contact with the drive rod. The extended model accurately reproduces the non-linear velocity decrease under load which is to be attributed to the prevailing motor operation in the slipping regime. The nature of the frequency-dependent stall-force limits, although well reproduced by the model, is more difficult to explain. Hess and Soom [99], in their studies on the dynamic behavior of friction, show that the friction force is lower for decreasing than for increasing velocities. This leads to a hysteresis loop in friction force with varying velocity. The loop becomes wider at higher rates of velocity change. This effect corresponds to energy loss which is more severe for high driving frequencies and is reproduced well by the LuGre model. The macroscopic effect of decreasing stall-force limits in the walking motor could be attributed to this phenomenon.
Beside reproducing the non-linear phenomena in motor operation, the analytic model also sheds light on other aspects affecting the perfor-mance of the motor. These include the resonant effects above 3kHz drive frequency and the relationship between the shape of the driving signals and the maximal level of motor preload. The new insights can be utilized in order to develop an alternative motor-drive strategy beyond the region of rated operation (see next section) and improve the force generation characteristics of the motor. Furthermore, the analytic modeling strategy resulted in a collection of linear subsys-tems not exceeding second order with a clear indication of non-linear influences. The modular structure of the overall model allows an easy extension of the model to cover additional aspects of motor operation (cf. sect.3.6), especially if low-level experimental data were provided.
Finally, the reproduction of low-level interactions between the legs and the drive rod in the physical model allows for its application in the optimization of driving signals [146] and the investigation of the feasibility of a biologically inspired drive approach (see below).
The second motor model derived in this work (see chapt.4) is based on the experimental approach. Although the analytic model accurately describes the dynamics of the walking motor and the non-linear phenomena in motor operation, its disadvantage lies in its complexity and non-linear nature. This renders its online application in the prediction of motor response and in the design of a suitable force control strategy difficult. The experimental model is meant to address these issues. For this reason, several simplifying assumptions were made. First, only the holistic motor behavior in xdimension is considered. Second, the experimental model is valid only for one particular driving waveform (force). Third, the model is constrained to drive frequencies up to2kHz. These simplifications result in a linear model as long as a constant load is applied to the motor. However, since the main purpose of this model is the design of a linear force controller for the application in a force control scenario, the load cannot be assumed constant. The experimental model addresses this issue by including the non-linear load-velocity dependency with help of a frequency-normalized polynomial fit on the experimental data.
The final model is non-linear but its linearity is eventually restored in chapt.7 with help of a feedback linearization technique in the context of force controller design. Although the generality of the analytic model is lost, the experimental approach is well suited to describe the dynamics of the motor in most practical scenarios. The final model accurately describes motor behavior for drive frequencies below2kHz and load levels below10N. Finally, the simple structure of the experimental model allows for its application in the design of a linear force controller in chapt.7.
Feasibility of a bioinspired drive approach
The second objective of this thesis was to investigate the feasibility of a bioinspired drive approach based on leg coordination mechanisms found in insects. This objective is motivated by the fact, that the motor in its current form can only be driven according to the walking principle (see sect.2.4.1) in which the legs move in pairs receiving the same driving signals. This drive strategy is hard-wired in the motor and is supposed to ascertain a stable operation of the motor by always providing the drive rod with two supporting contacts to the legs. However, two reasons speak against this strategy. First, if more legs were allowed to contact the drive rod, the force generation capacity of the motor could be improved. This goal harmonizes with the application of the walking motor as a force generator followed in this work. Second, the waveform optimization strategy proposed
by Merry et al. [146] indirectly shows that even with a considerable flexibility in the design of the shape of the driving waveforms, only small improvements in the performance of the motor are possible.
Thus Merry’s approach can be seen as an evidence of the inherent limitations of the pairwise drive strategy. If a coordination mechanism existed which would ascertain a stable operation of the motor (i.e. at least two leg contacts at any given time) but would not rely on fixed pairwise relations among the legs, more legs could contact the drive rod resulting in a possibly stronger motor. This thesis has shown that such a mechanism exists and that the performance of the motor can be significantly improved not only in terms of force generation (up to50% higher stall force limits) but also in terms of maximal motor velocity (up to 100% higher velocity, see chapt.5). The proposed bioinspired solution relies on leg coordination mechanisms found in stick insects by Cruse et al. [47,49]. Beside the idea of a non-standard application of the biological findings, the contribution of this work lies in a successful architectural mapping between the different mor-phologies of the original six-legged model and the walking motor.
Moreover, a new strategy for waveform generation has been proposed which is intuitive and respects the admissible work area of the legs.
The evaluation of the bioinspired approach would not be possible without the physical model of motor dynamics developed in chapt.3 of this thesis. In fact, the superiority of the bioinspired approach could only be shown in computer simulation since the real motor is hard-wired for the pairwise drive strategy. The necessary adjust-ments in the motor allowing the independent operation of all legs are minimal. The only component which has to be modified is the flex circuit connecting the external electrical phases to the legs of the motor. Unfortunately, this simple adjustment has to be done during the manufacturing process and was not possible for the author. Never-theless, the theoretical results obtained in this work are reliable. This is motivated as follows. First, the biologically inspired coordination mechanism guarantees the stable operation of the motor. This can be easily seen by considering the number of legs contacting the drive rod at any time which is always greater than or equal two. Second, for low and moderate drive frequencies there are at least three or four legs contacting the drive rod most of the time which has to improve the load characteristics of the motor. Third, the shape of the waveforms can be varied flexibly which in the combination with independent control of the legs results in the largest possible waveform design flexibility. This fact should again be compared to the work by Merry et al. [146] since the motor model proposed by Merry, and used to evaluate the waveforms designed for the pairwise drive strategy, is less general than the physical motor model developed in this work.
However, even with the less general model, Merry could show the accordance between the simulation and real experiment for the newly
designed waveforms. The new degree of freedom gained through the independent control of all legs can hardly result in a worse per-formance of the motor. In fact, the optimization process has shown that the performance of the motor improves rapidly even for a simple choice of the objective function. Considering the fact, that only two optimization criteria were pursued in this work – both resulting in a significant improvement in the performance of the motor – and that only one particular architectural mapping between the motor and the biological model with one particular selection of coordination rules were investigated, the proposed approach has the potential to further improve other aspects of motor performance.
Development of a force control strategy
The third objective of this work was the development of a force control strategy suitable for the application in a biologically inspired robot joint. In this context, two goals were followed. First, the experimen-tal motor model from chapt.4was to be linearized in order to take advantage of the rich repertoire of mathematical tools for the design of a linear controller. Second, the controller to be developed was to consider the influence of series elasticity on force transmission.
The first goal was addressed by developing a load-compensation strategy based on force feedback in chapt.7. The proposed strategy restores the linearity of motor operation even under load if moderate drive velocities up to1kHz and load levels up to5N are not exceeded.
The experiments with active load compensation have shown that even though the effect of external force cannot be compensated completely for the whole operating range, the compensation keeps the motor velocity at a constant level for moderate drive frequencies and force levels. The above limits apply since the drive frequency of the motor cannot grow infinitely in order to maintain the desired drive velocity.
Furthermore, for high levels of load the non-linear effects in motor op-eration due to friction gain in importance and cannot be compensated easily. For the moderate levels of load force, the compensated motor model can be considered linear.
The fulfillment of the second goal – design of a force controller – was based on the linearized motor model with load compensation.
Although model based approaches were applicable, the actually de-signed controller is of PI type due to its better robustness against model uncertainties. The influence of series elasticities on force trans-mission was considered through the development of a sensor-tendon model incorporating the dynamics of the force sensor in series with an elastic tendon modeled as linear spring of a given stiffness. The designed controller was tested in the simulation and in real-world ex-periments by pulling on tendons of different elasticities and showing a good agreement between the model and the reality. It has been shown that the designed controller can be successfully applied in a force
con-trol scenario as long as the series elasticity remains at the effective level of 10 N/mm or above and the rate of change in the reference force does not exceed 10 Hz. It should be noted, that depending on the point of view, the designed force controller is either a linear PI type acting on a linearized motor or a non-linear adaptive controller incorporating the load compensation strategy acting on a non-linear motor. As an additional contribution of this work, the limits on the performance of a perfect force controller have been investigated theoretically in dependency of the effective stiffness of the force transmitting tendons.
The performance of the designed controller lies at the level of55% of the theoretical limit.
Feasibility of a muscle-like force generation
The fourth objective of this work was to test the feasibility of a bioin-spired application of the walking motor as a force generator in a small-sized robot joint. Two motors were supposed to actuate an antagonistic joint by transmitting pulling forces through tendons ac-cording to the concept in Fig.3. The approach is bioinspired due to the antagonistic arrangement of the actuators and due to the idea that the motor together with a force controller and a suitable sensory feed-back can mimic the force generation characteristics of a muscle. The long-term objective of such an approach is the possibility to control the technical actuators by means of myoelectric activity in prosthetic devices. Accordingly, a simple 1-DOF joint was built allowing for an antagonistic arrangement of the motors and the integration of po-sition and force sensors as the pre-requirement for muscle mimicry based on the model by Hill (see chapt.8). Additionally, in order to prevent tendons from going slack the force control architecture from chapt.7 was extended with the concept of a virtual tendon. The force controllers were supposed to track reference forces according to the force-length relationship of a muscle for the given geometry of the joint, position of the “virtual muscles” and the levels of their activation. The final mechanical setup together with motor-drive elec-tronics and control algorithms was tested in a simple position control scenario. Beside the audible operation of the motors, which could possibly be alleviated through damming in a practical application, the feasibility of the proposed strategy was confirmed. The joint could track the commanded positions with the largest overshoot below5% of the reference step signal which is a good result considering the complexity of the approach chosen. In fact, with the given technical system a positioning task could have been achieved in a better way by employing a direct position control without muscle mimetic. How-ever, the strength of the proposed approach lies in the possibilities of its extension with regard to future applications and neurobionic control strategies. The contribution of this work lies in the novelty of the presented approach since piezoelectric motors have not been
employed as technical muscles or even force generators in any other work known to the author. This work confirms the feasibility of such application. The feasibility of muscle mimetic by means of control of technical actuators in general is a much broader question beyond the scope of this thesis. Presumably, a pure control-based approach without the integration of real elastic elements is not sufficient since no controller is fast enough to counteract a shock. Moreover, the exact role of joint geometry, co-activation of antagonistic muscles, non-linear muscle characteristics and reflexes in the generation of movement is still not sufficiently understood. Some of these open questions have been addressed recently in [5, 6]. The simplistic architecture with
“virtual muscles” as presented in this work is a framework for further investigation of these topics.
Beside the above contributions, a small-sized motor-drive electron-ics as introduced in chapt.6has been developed within the scope of this work and the diploma thesis of Daniel Basa [14] as well as the Bachelor thesis of Tim Walther [219]. The newly developed electronics supersedes the commercially-available products due to its compact-ness and the possibility of waveform generation at much higher drive frequencies, above50kHz, as compared to commercial products. The latter feature is the foundation for the development of an alternative motor-drive strategy in overdrive mode (see next section). The circuit diagrams and the PCB layout images of the new electronics are in-cluded in appendixC.
Other minor contributions of this work are the development of an algo-rithmic and a practical approach for waveform generation at a desired drive frequency and of a motor direction switching strategy which is compatible with the bioinspired waveform generation approach (see chapt.6).