Intelligent proportional-integral (iPI) control of a single link flexible joint manipulator

Abstract

This paper presents the design, stability analysis and experimental validation of a computationally non-intensive, model-free, intelligent proportional-integral (iPI) controller for flexible joint manipulators. In order to show the performance of the iPI controller, it is compared with classical proportional-integral and proportional-integral-derivative controllers. Based on this comparison, the iPI-controlled system achieved a better than 60% tracking accuracy for both kane trajectory and sine input tracking. The iPI controller also significantly reduced transient swings in the flexible joint of the manipulator, when tracking a train of pulses. Moreover, the iPI controlled system successfully eliminated both disturbances and noise effects from the dynamics of the manipulator.

Keywords

Flexible-joint manipulator intelligent PI control PI control PID control trajectory tracking

1. Introduction

The high energy consumption, relatively low speeds of motion, and high manufacturing and actuator costs of rigid-body robotics motivated the exploration of flexible robotic structures. Along with several benefits and applications of flexible joint mechanisms, the joint elasticity leads to several control challenges. Nonlinearities, model uncertainties, friction, disturbances and noise effects complicate model-based controller design in flexible-joint manipulators. Attempts at solving the dynamical challenges of joint-flexibility have resulted in several control strategies being applied for the control of these systems. Input command shaping, for example, has been used in the control of residual vibration in this class of manipulators (Spong, 1989; Ott et al., 2004; Kapucu et al., 2006; Yavuz et al., 2011), but many of these vibration control strategies are essentially open-loop based. Several model-based control strategies including variable structure control, feedback linearization and backstepping control have been used (Marino and Spang, 1986; Sira-Ramirez and Spong, 1988; Sira-Ramirez and Zribi, 1992; Sira-Ramirez and Agrawal, 2004; Khorasani, 1990; Brogliato et al., 1995; Oht and Leet, 1997; Vakil et al., 2011): model uncertainties, frictional effects and disturbances limit the effectiveness of these model-based control strategies. Robust and adaptive variants of such model-based controllers have also been reported to improve controller performance in the presence of model uncertainties, parameter variations, disturbances and frictional effects (Ghorbel et al., 1989; Han and Chen, 1993; Ge, 1996; Kim and Oh, 2006; Mamani et al., 2011; Zhao et al., 2011), but these add significantly to computational and implementation burdens. Model predictive control strategies have also been employed, with the potential to reduce the computational burden associated with adaptive control strategies (Kandroodi et al., 2012; Zhan and Hu, 2012). Recent departures from model-based control strategies for this class of systems use artificial-intelligence (AI)-based systems (Passino and Yurkovich, 1995; Wong et al., 1995; Jang et al., 1997; Subudhi and Morris, 2003; Siddique and Tokhi, 2006; Renno, 2007; Chaoui and Gueaieb, 2008; Maouche and Attari, 2008; d’Andrea-Novel et al., 2010; Akyuz et al., 2011, 2012). While several results have been reported from both theoretical and experimental studies, the design of these controllers requires specialist knowledge. Moreover, the computational costs of tuning the most innovative of these AI controllers are significant.

A more recent development with the potential to design controllers for systems with significant model uncertainty and disturbances, using minimum specialist skills, is the use of iPI and intelligent proportional-integral-derivative (iPID) controllers (Fliess and Join, 2008, 2009; Fliess et al., 2011). Also, an application of intelligent controllers in a multilink system can be found in Fliess et al. (2011). This paper presents the implementation of an iPI controller for the control of a single-link flexible joint robot. In the rest of the paper, a recollection of the model of the flexible- joint robot is presented in Section 2: the presentation highlights the challenges of model-based design and motivates the model-free approach used in this paper. Controller design, real-time controller implementation and experimental results are given in Section 3 and Section 4, respectively. Results and discussions are described in Section 5. Finally, conclusions are summarized in Section 6.

2. Modeling of the flexible-joint manipulator

A solid model of the manipulator employed for this study is shown in Figure 1. Figure 2 shows a photograph of the actual experimental setup, which consists of a dspace ds1103 real-time controller, a PC, a geared brushless DC motor with encoder (Faulhaber 2444S), a motor driver (Faulhaber MCBL5004), incremental encoder (2000 P/R), springs (58.86 N/m) and rigid link (0.4 m). This system is modeled (Kandroodi et al., 2012) by equation (1):

J_{t} \overset{··}{θ} + I (\overset{··}{θ} + \overset{··}{α}) + B \overset{\cdot}{θ} = u (t)

(1a)

I (\overset{··}{θ} + \overset{··}{α}) + k_{s} α = 0

(1b)

u (t) = \frac{η_{m} η_{g} K_{t} K_{g}}{R_{m}} v (t) - \frac{η_{m} η_{g} K_{t} K_{m} K_{g}^{2}}{R_{m}} \overset{\cdot}{θ}

(1c)

where: I is the inertial of the link of length (l); J_t is the total inertia of the system referred to the rotor of the drive; m is the mass of the payload; R_m is the motor armature resistance; K_g is the gear ratio; K_m is the back emf of the motor and

η_{m}, η_{g}

are the efficiencies of the motor and gear train respectively. u(t), v(t), θ and α are the drive torque, the motor voltage, the motor rotation angle and the link deflection angle, respectively.

2.1. Differential flatness of the manipulator and the existence of an open-loop pole at the origin

Using the concepts of differential flatness, the manipulator model in equation (1) can be recast in the form of equation (2). This requires the derivation of a flat output, y(t); such that all the states of the flexible-joint manipulator, together with its input, could be expressed in terms of the flat output and a finite number of its derivatives. The detailed procedure for the derivation of the flat output for a linear system can be found in Levine and Nguyen (2003). The flat output for the manipulator, together with its first two derivatives, is given in equation (2a):

y = θ + α, \overset{\cdot}{y} = \overset{\cdot}{θ} + \overset{\cdot}{α}, \overset{··}{y} = \overset{··}{θ} + \overset{··}{α}

(2a)

α = - \frac{I}{K_{s}} \overset{··}{y}, \overset{\cdot}{α} = - \frac{I}{K_{s}} y^{(3)}, \overset{··}{α} = - \frac{I}{K_{s}} y^{(4)}

(2b)

θ = y - α = y + \frac{I}{K_{s}} \overset{··}{y}, \overset{\cdot}{θ} = \overset{\cdot}{y} + \frac{I}{K_{s}} y^{(3)}, \overset{··}{θ} = \overset{··}{y} + \frac{I}{K_{s}} y^{(4)}

(2c)

E (y, {\overset{\cdot}{y}}_{1}, \overset{··}{y}, y^{(3)}, y^{(4)}, v (t)) = \frac{J_{t} I}{K_{s}} y^{(4)} + \frac{a_{1} I}{K_{s}} y^{(3)} + (J_{t} + I) \overset{··}{y} + a_{1} \overset{\cdot}{y} - a_{2} v (t) = 0

(2d)

where

a_{1} = ({BR}_{m} + η_{m} η_{g} K_{t} K_{m} K_{g}^{2}) / R_{m} and a_{2} = (η_{m} η_{g} K_{t} K_{g}) / R_{m} .

Equation (2a) follows from the definition of the flat output; equation (2b) and (2 c) is obtained by substitution of the flat output and its appropriate derivatives into equation (1b). Equation (2d) is obtained by substituting equation 2(b) into equation (1a).

It follows from equation (2d) that the transfer function of the system in terms of the flat output is given by:

G (s) = \frac{Y (s)}{V (s)} = \frac{a_{2}}{(J_{t} I / K_{s}) s^{4} + (a_{1} I / K_{s}) s^{3} + (J_{t} + I) s^{2} + a_{1} s}

(3)

It is evident from equation (3) that the system has an open-loop pole at the origin of the s-plane. This leads to the integrator-type instability shown in Figure 3.

2.2. Model uncertainties

The parameter subset Ω = [I, M, k, K_m, K_g, R_m, η_g, η_m]^T, of the system is not completely known, since m and J_t vary with the payload; and the joint constant K_s and motor parameters are affected by ageing. In addition, both the motor efficiency η_m and the gear efficiency η_g will vary with time, and with operating conditions. In part, this has been the basis for most of the robust and adaptive control strategies used for the control of this class of manipulators (Han and Chen, 1993).

2.3. Frictional effects and unmodeled disturbances

The model in equation (2) does not include the frictional effects that modify the system dynamics. Including these effects and disturbances modifies the manipulator model, yielding equation (4):

E (y, \overset{\cdot}{y}, \dots, y^{(4)}, μ (I, J, m, K_{g}, R_{m}, l, K, K_{m}), f_{1} (\overset{\cdot}{α}), f_{2} (\overset{\cdot}{θ}), d (t), v (t)) = 0

(4a)

| d (t) |_{max} = d_{max} \leq \infty

(4b)

In line with the model-free approach of the current paper, there is an exact description of the two friction terms

f_{1} (\overset{\cdot}{α}), f_{2} (\overset{\cdot}{θ})

, which are a combination of average defection of the elastic bristle, stiffness of the elastic bristle, micro-damping, viscous damping, Coulomb friction and stiction level (Zhao et al., 2011). Model uncertainties, unknown disturbances and nonlinear frictional effects combine to compromise performance of model-based controllers. Consequently, model-based controllers require substantial online adaptation to sustain an acceptable performance (Ghorbel et al., 1989; Han and Chen, 1993; Zhao et al., 2011). Moreover, it is these modeling difficulties that motivate the exploitation of model-free control strategies in the control of the flexible-joint manipulator.

3. Design of the iPI controller

Recall equation (4), which reflects the effects of friction and parameter uncertainties in the manipulator model. In the controller design approach presented in this paper, the implicit function theorem allows an ultra-local representation of the system in equation (4) to be written, valid for a very short time, as in equation (5):

ψ (y, \overset{\cdot}{y}, \dots, y^{(j)}, v (t)) = 0

(5)

where v(t) and y are the input and output variables respectively; and ψ is a sufficiently smooth function of order j. For the design of the so-called intelligent controllers, the ultra-local model admits the representation in (5).

y^{(n)} = Φ (y, \overset{\cdot}{y}, \dots, y^{(l)}, v (t)) = 0; n = 1, 2

(6)

For the intelligent proportional integral (PI) controller design (Fliess and Join, 2008, 2009; Fliess et al., 2011), n = 1, such that:

\overset{\cdot}{y} = F (t) + α v (t)

(7)

The implementation procedure for this controller (Fliess and Join, 2009; Fliess et al., 2011) is given by equation (8).

v_{c} (t) = - \frac{F}{α} + \frac{\overset{\cdot}{y}}{α} + K_{p} e + K_{I} \int edt

(8)

where

y^{*} (t)

is the reference trajectory, being tracked by the controlled manipulator and e is the trajectory error (

e = y^{*} (t) - y (t)

). In the implementation, the structure of the controller was slightly modified as follows:

Select the constant input parameter α

Then compute or update controller parameters as:

\overset{\land}{F} (t) = \overset{\cdot}{y} (t) - α v_{c} (t - δ); F_{i = 0} = 0

(9a)

v_{c} (t) = - \frac{\overset{\land}{F} (t)}{α} + \frac{{\overset{\cdot}{y}}_{r} (t)}{α} + K_{p} (y_{r} (t) - y (t)) + K_{I} \int (y_{r} (t) - y (t)) dt

(9b)

where

\hat{F} (t)

is the current estimate of F(t), using

\overset{\cdot}{y} (t)

, the first derivative of the current output measurement.

v_{c} (t)

is the current output of the controller, computed using the current estimate of F(t), the current value of

{\overset{\cdot}{y}}_{r} (t)

the first derivative of the reference trajectory, the previous input

v_{c} (t - δ)

and the supplementary PI signal.

3.1. Stability analysis of the iPI controller

For the stability analysis of the controller, substitute equation (9a) into 9(b) to obtain:

v_{c} (t) = - \frac{\overset{\land}{F} (t)}{α} + \frac{\overset{\cdot}{y} (t)}{α} + K_{p} (y_{r} (t) - y (t)) + K_{I} \int (y_{r} (t) - y (t)) d t = - \frac{1}{α} \overset{\cdot}{y} (t) + v_{c} (t - δ) + \frac{1}{α} {\overset{\cdot}{y}}_{r} + K_{p} (y_{r} (t) - y (t)) + K_{I} \int (y_{r} (t) - y (t)) d t

(10)

Taking Laplace transform of the equation above, it is obtained as follows:

α v_{c} (s) (1 - e^{- δ s}) = (Y_{r} (s) - Y (s) (s + α K_{p} s + \frac{α}{s} K_{I})

(11)

where α and the controller gains in equation (11) are selected such that the polynomial

(s^{2} + α K_{p} s + α K_{i})

is Hurtwiz. For a time delay transfer function, a first-order approximation is used as

e^{- δ s} \approx (1 - (δ / 2) s) / (1 + (δ / 2) s)

, then transfer function of the controller is obtained as in equation (12).

G_{c} (s) = \frac{V_{c} (s)}{E (s)} = \frac{(1 + \frac{δ}{2} s) (s^{2} + α K_{p} s + α K_{I})}{α δ s^{2}}

(12)

where

E (s) = Y_{r} (s) - Y (s)

. It follows from equations (3) and (12), that the closed-loop transfer function of the iPI controlled manipulator has the following characteristic equation:

Q (s) = α δ s^{2} (\frac{J_{t} I}{K_{s}} s^{4} + \frac{a_{1} I}{K_{s}} s^{3} + (J_{t} + I) s^{2} + a_{1} s) + a_{1} (1 + \frac{δ}{2} s) (s^{2} + α K_{p} s + α K_{I})

(13)

Three cases of stability could be deduced from equation (13).

Case I

If either δ→0 or $| z | << \frac{1}{\sqrt{α δ}}$ , then the characteristic roots of the closed-loop system under iPI control are approximated by the zeros of the polynomial $R_{1} (z) \approx a_{1} (1 + \frac{δ}{2} z) (z^{2} + α K_{p} z + α K_{i})$ : then, the closed-loop dynamics of the manipulator has third-order dynamics, which could be assigned independently of the manipulator parameters: or model-free control is achieved. The controller parameters can be tuned using the Routh-Hurwitz criterion. It is therefore evident that the computational burden of tuning the iPI controller is significantly lower than that due to alternative linear, nonlinear and AI-based controllers for the same system.

Note particularly, when $δ \to 0$ , the dynamics of the third pole, $z = - 1 / δ \to - \infty$ , decay rapidly, such that the closed-loop dynamics of the iPI controlled manipulator further reduces to second-order, with the dominant pole-pair given $s_{1}, s_{2} = (- α K_{p} \pm \sqrt{α^{2} K_{p}^{2} - 4 α K_{I}}) / 2$ .

Case II

If $α δ | z |^{2} \to \infty$ , then the closed-loop dynamics are approximated by $R_{2} (z) \approx α δ z^{3} ((J_{t} I / K_{s}) z^{3} + (a_{1} I / K_{s}) z^{2} + (J_{t} + I) z + a_{1})$ and the system has three closed-loop poles at the origin of the s-plane, which is unstable.

Case III

The general situation where

Q (s) = \frac{α δ J_{t} I}{K_{s}} s^{6} + \frac{α δ a_{1} I}{K_{s}} s^{5} + α δ (J_{t} + I) s^{4} + (α δ a_{1} + \frac{δ}{2}) s^{3} + [1 + \frac{α δ K_{P}}{2}] s^{2} + [α K_{p} + \frac{α δ K_{I}}{2}] s + α K_{I} = 0

(14)

requires a parameter-based tuning of the system, which is not the subject of the current discourse.

3.2. Real-time controller implementation using the dSPACE rapid prototyping system

The real-time implementation of the controllers was undertaken using dSPACE DS1103 based on Simulink. DS1103 is a powerful real-time controller that was developed especially for rapid prototyping purposes. Figure 4 shows the implementation arrangement in Simulink.

The Simulink model for the real-time implementation of the iPI controller is shown in Figure 5. In order to compare the iPI controller with other controllers, the classical PI and PID controllers were also designed. Hence, the Simulink model allows any one of the three controllers to switch-in according to the conditions given in Table 1.

Table 1.

Controller switching table.

Controller	Intelligent gain	K_p	K_i	K_d
iPI	1	1	0.01	0
PI	0	1	0.01	0
PID	0	1	0.01	0.01

iPI: intelligent proportional integral; PI: proportional integral; PID: proportional integral derivative.

4. Experimental results

Three controllers were experimented with: the classical PI, classical PID and the iPI controllers, respectively. First of all, the classical PI controller was tuned on the real-time implementation system until an acceptable response of the flexible manipulator was realized. The PI controller parameters were selected as K_P = 1, K_i = 0.01 (these K_p and K_i were, in fact, used in all three controllers). Then, the derivative gain of the classical PID was obtained as K_D = 0.01. The following experimental investigations were undertaken.

4.1. Effects of variation of α on the performance on the iPI controller

The iPI controller was tested for several values of the input parameter α. Sample results of the effects of α on F(t), e(t), and the output y(t), of the flexible-joint manipulator are shown in Figure 6.

The best value for α was found to be 0.4 and it was used for the rest of the experimental work. This value is of an order similar to values used in other studies (Fliess and Join, 2009).

4.2. Trajectory tracking

Traditionally, most controllers for flexible joint robots employ trajectories of motion. Point to point and continuous trajectory tracking experiments were applied to each of the three controllers. The trajectories were generated using Kane’s transition function, which is given in equation below.

y (t) = y_{0} + (y_{f} - y_{0}) \frac{t - t_{0}}{t_{f} - t_{0}} - \frac{y_{f} - y_{0}}{2 π} \sin (2 π \frac{t - t_{0}}{t_{f} - t_{0}})

(15)

where y(t) is the generated trajectory, y₀ and y_f are the start and final position, t is time, t₀ and t_f are the start and final time. The trajectory results related to Kane’s transition function are shown in Figure 7.

4.3. Dynamic response of manipulator

The dynamic response of the manipulator to a step input, a sine input and a train of square pulses, respectively, was investigated for each of the three controllers. The results are illustrated in Figures 8–10.

4.4. Disturbance rejection

The manipulator was steered to an initial position of θ = 0, α = 0, to eliminate the impact of velocity-dependent frictional effects on the dynamics of the system. An impulse mechanical disturbance was then applied to the manipulator. The disturbance rejection capability of the three controllers is shown in Figure 11.

4.5. Controller performance under noisy measurement

The performance of the iPI, the classical PI and the PID controllers in the presence of ±0.2° uniformly distributed white measurement noise is presented through the system responses shown in Figure 12. An example view of the noise signal is also shown in Figure 12.

5. Results and discussion

It is evident from Figure 6, that the impact of α on the dynamics of the manipulator cannot be uniquely assessed. This is consistent with suggestions in the literature (Fliess and Join, 2008, 2009; Fliess et al., 2011), which say that the physical significance of this parameter cannot be exactly quantified. Whereas the kane trajectory tracking results are shown in Figure 7, Table 2 summarizes the comparative performance of the three controllers, based on the root-mean squared evaluation of the tracking error and control signals. The classical PI controller exhibited the highest trajectory tracking error. The iPI controller yielded better tracking accuracy (59%) when the control signals had almost the same energy.

Table 2.

Root-mean-square (RMS) error during Kane trajectory tracking.

Controller	RMS Control Effort	RMS Tracking Error	Comparative error percentage
iPI	0.0102	6.73E-3	59%
PI	0.01	1.64E-2	0%
PID	0.0097	1.59E-2	3.1%

iPI: intelligent proportional integral; PI: proportional integral; PID: proportional integral derivative.

Please refer to both Figures 8–10 and Tables 3–5. The overall error in the step response is essentially of the same magnitude, for all of the three controllers. Rising times of the controllers were almost the same (3.5 sec) and iPI showed no α swing effect. On the other hand, its settling time was slower than the others. The worst overshoot of 21% was obtained under the classical PI controller, when compared to the overshoot of 18.7% and 19.3% under the iPI and PID, respectively. It is evident from Figure 9 that the iPI control led to a time lag of 0.06 seconds of the output as compared to the input sine wave. This is better than the 0.128 second lag-time resulting from the two classical controllers. This causes better (60%) sine input tracking accuracy under iPI control, as recoded in Table 4. Moreover, iPI control results in a better damping of swings in the flexible joint, as evidenced by the better filtering of the α-component of motion during square-wave tracking, as recorded in Figure 10.

Table 3.

Comparison of root-mean-square (RMS) error under step input tracking.

Controller	RMS Control Effort	RMS Tracking Error	Comparative error percentage
iPI	0.0408	0.662	0%
PI	0.0297	0.634	4.2%
PID	0.0289	0.629	5%

iPI: intelligent proportional integral; PI: proportional integral; PID: proportional integral derivative.

Table 4.

Comparison of root-mean-square (RMS) error to tracking of sine input.

Controller	RMS Control Effort	RMS Tracking Error	Comparative error percentage
iPI	0.0698	0.0465	60.1%
PI	0.0699	0.1164	0%
PID	0.0666	0.111	4.6%

iPI: intelligent proportional integral; PI: proportional integral; PID: proportional integral derivative.

Table 5.

Comparison of root-mean-square (RMS) error for tracking a square wave.

Controller	RMS Control Effort	RMS Error	Comparative error percentage
iPI	0.0693	0.1143	0%
PI	0.0471	0.1044	8.7%
PID	0.0479	0.1064	6.9%

iPI: intelligent proportional integral; PI: proportional integral; PID: proportional integral derivative.

As can be seen from Figure 11, all of the three controllers eliminated the disturbance effect within 1 second. The iPI controller yields less over/undershoot than the other controllers. The performance of the three controllers under white noise effects is shown in Figure 12. The tracking error of the classical controllers is more than double that due to the iPI control. Moreover, the iPI controller sustains adequate trajectory tracking throughout the period of observation. On the other hand, the classical controllers track the reference trajectory with some error during the dynamic period of the trajectory.

6. Conclusion

In this paper, the iPI controller was developed for a single link flexible joint manipulator. In order to test the robustness and trajectory performance of the controller, several experiments were conducted. Also, the performance of the iPI controller was compared with the classical PI and PID controllers. The iPI controller used in the flexible joint system yielded better performance during the tracking of arbitrary point-to-point trajectories and sine inputs. Better control of the joint motion of the manipulator under step and square wave inputs were observed under iPI control. Under the measurement noise, the iPI controller was also shown to perform better experimentally. Furthermore, iPI can handle the external disturbances successfully.

The model-free tuning of this type of controller makes it particularly suitable for use in the flexible joint manipulator where nonlinearities, payload changes and velocity dependent regimes of friction lead to model changes. This model-free approach makes the controller suitable for use where parameter changes due to field exposure and ageing can result in model uncertainty. The low computational burden required in the design and implementation of this class of controllers, as opposed to the computational intensity and specialist skills required in the design and implementation of alternative model-free controllers based on artificial intelligence schemes, give iPI controllers a comparative advantage. There is, therefore, significant motivation for the further exploration of iPI controllers in practical applications of flexible joint manipulators. In particular, a dynamic model of multiple-link manipulators is very complicated and it has a lot of nonlinear terms in mass and damping matrices. The iPI controller has great potential for the control of multi-link manipulators,

Footnotes

Acknowledgements

Professor John T Agee wishes to gratefully acknowledge the financial support received from the National Research Fund (NRF) of South Africa.

Funding

This work was supported by the National Research Fund (NRF) of South Africa (through the Equipment Mobility grant award number NRF UID no: 80475), which facilitated this research activity.

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