State-dependent control of a hydraulically actuated nuclear decommissioning robot

Control Engineering Practice 21 (2013) 1716–1725

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State-dependent control of a hydraulically actuated nuclear decommissioning robot C. James Taylor n, David Robertson Engineering Department, Lancaster University, LA1 4YR, UK

art ic l e i nf o

a b s t r a c t

Article history: Received 4 December 2012 Accepted 19 August 2013 Available online 26 September 2013

This paper develops and evaluates state-dependent parameter (SDP) control systems for the hydraulically actuated dual-manipulators of a mobile nuclear decommissioning robot. A unified framework for calibration and SDP model identification is proposed, in which the state-dependent variable is a delayed voltage input associated with the time-varying system gain. Such nonlinearities can cause undesirable joint movements under automatic control. Hence, the present paper develops a nonlinear pole assignment algorithm for the SDP model. Closed-loop experimental data shows that the SDP design more closely follows the joint angle commands than an equivalent linear algorithm, offering improved resolved motion. & 2013 Elsevier Ltd. All rights reserved.

Keywords: State-dependent parameter Non-minimal state space Pole assignment Nuclear decommissioning robot Hydraulic manipulators

1. Introduction With rapid start and stop, fast responses in general, and large torque-to-weight ratios, hydraulic robots are suitable for many applications. They are commonly employed by the construction, mining and nuclear industries, where semi-automatic control systems are being adopted as a means of improving the efficiency, quality and safety of operations. One challenge for system developers is the achievement of fast, accurate movement of manipulators under automatic control (Mohanty & Yao, 2011a, 2011b; Sirouspour & Salcudean, 2001; Tafazoli, Salcudean, Hashtrudi-Zaad, & Lawrence, 2002). The problem is made difficult by a range of factors that include highly varying loads, speeds and geometries. In the civil and construction industries, automated prototypes include, for example, hydraulic manipulators for site assembly of block walls (Bock, Stricker, Fliedner, & Huynh, 1996), ground compaction (Shaban, Ako, Taylor, & Seward, 2008) and, most notably, hydraulic excavators (Bradley & Seward, 1998; Budny, Chlosta, & Gutkowski, 2003; Gu, Taylor, & Seward, 2004; Ha, Nguyen, Rye, & Durrant-Whyte, 2000; Lee & Chang, 2002). In the nuclear industry, where the use of people in areas of significant contamination is not always possible, mobile hydraulic robots provide an invaluable option for the safe retrieval of contaminated materials, whilst safeguarding the environment and minimizing radiation exposure to operators (Bakari, Zeid, & Seward, 2007;

n

Corresponding author. Tel.: þ 44 1524592375. E-mail addresses: [email protected] (C.J. Taylor), [email protected] (D. Robertson). URL: http://www.lancs.ac.uk/staff/taylorcj/ (C.J. Taylor). 0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2013.08.011

Iborra, Pastor, Alvarez, Fernandez, & Merono, 2003; Seward, Pace, & Agate, 2007). In contrast to a typical machine driven by electric motors, hydraulic actuators generally have higher loop gains, wider bandwidths and lightly damped, nonlinear dynamics (Merritt, 1976). In general, research into nonlinear control of robotic manipulators embraces numerous approaches, such as adaptive (Guan & Pan, 2008), sliding mode (Feng, Yu, & Man, 2002), fractional robust (Pommier, Sabatier, Lanusse, & Oustalop, 2002), exact linearization by feedback (Kreutz, 1989), various Lyapunov-based methods (Alleyne & Liu, 2000; Su, Muller, & Zheng, 2010) and linear parameter varying (LPV) control (Hashemi, Abbas, & Werner, 2012). LPV and the related state-dependent parameter (SDP) approaches are based on the definition of quasi-linear models. The parameters of SDP models are functionally dependent on measured variables (Young, 2001), such as joint angles and velocities in the case of manipulators (Taylor, Shaban, Stables, & Ako, 2007). Although relatively few papers concentrate on control design specifically for hydraulic manipulators, selected examples consider backstepping (Sirouspour & Salcudean, 2001; Zeng & Sepehri , 2005), impedance (Ha et al., 2000; Tafazoli et al., 2002), adaptive robust (Guan & Pan, 2008; Mohanty & Yao, 2011a), decentralized time-delay (Lee & Chang, 2002), generalized predictive (Kotzev, Cherchas, Lawrencet, & Sepehrit, 1992), variable structure (Jerouane, Sepehei, & Lamnabhi-Lagarrigue, 2004) and sliding mode (Indrawanto, 2011) methods. Particular challenges associated with hydraulic systems include their friction characteristics, valve saturations and deadbands (Mohanty & Yao, 2011a), hydraulic fluid compressibility (Sirouspour & Salcudean, 2001) and, as demonstrated in the present paper, asymmetric actuation (Alleyne & Liu, 2000). Liu and Daley (2000) use the inverse of the deadband

C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

in hydraulic systems, allowing for the design of an optimally tuned nonlinear PID controller. Model uncertainties encompass the accumulation of oil contamination, potential leakages in the hydraulic circuit (Mohanty & Yao, 2011a, 2011b) and the changing viscosity of hydraulic fluid due to temperature variations (Kotzev et al., 1992). In the context of nuclear decommissioning, it is also necessary to take into account the large variety of items that have to be dismantled and the geometric changes that occur during the dismantling process (Bakari et al., 2007). In the early stages of nuclear clean-up, for example, expensive, bespoke machines were designed, built and commissioned for specific projects. More recently, engineers have striven for more cost effective and flexible off-the-shelf systems but these can suffer from a converse restriction, lacking the ease of control afforded by high-specification custom designs (Taylor & Seward, 2010). Hence, the research described in this paper aims to develop generic control architectures that can be readily adapted to different tasks in the industry. More specifically, the experimental results in the paper are based on a dual-arm robotic platform, namely a Brokk-40 demolition robot with caterpillar tracks, to which two seven-function HydroLek-7W robotic manipulators have been attached (see e.g. Taylor, Chotai, & Robertson, 2010; Taylor & Seward, 2010). Here, it is desirable to develop relatively straightforward control algorithms that can be implemented using the existing commercial hardware and software framework, which was originally designed for conventional Proportional–Integral–Derivative (PID) methods. Unfortunately, devices initially developed for heavy lifting are not necessarily suitable for ‘soft touch’ duties such as picking up relatively fragile objects or for automatically aligning a tool with a workpiece. Indeed, for the reasons discussed above, notably the various system nonlinearities, the manipulators considered in this paper can sometimes suffer from a relatively slow and imprecise control action when using the existing linear PID feedback algorithms. Hence, to help improve control performance, earlier research using the Brokk-HydroLek system (Bakari et al., 2007; Bakari, 2008) considered model-based, Proportional–Integral–Plus (PIP) control systems (Taylor, Chotai, & Young, 2000; Young, Behzadi, Wang, & Chotai, 1987). The linear PIP controller can be interpreted as a logical extension of conventional PID design but with additional dynamic feedback and input compensators introduced automatically when the process has second order or higher dynamics, or pure time delays greater than one sampling interval. For such PIP design, linear (timeinvariant) Non-Minimal State Space (NMSS) models (Gonzalez, Perez, & Odloak, 2009; Taylor et al., 2000; Young et al., 1987) are first formulated, so that full state variable feedback control can be implemented directly from the measured input and output signals of the controlled process, without requiring the complexity of a deterministic state reconstructor or a stochastic Kalman filter. Among numerous other application areas (see e.g. Taylor, Young, & Chotai, 2013 and the references therein), linear NMSS-PIP control algorithms have previously been applied to mini-tracked hydraulic excavators (Gu et al., 2004) and hydraulic vibro-lance systems (Shaban et al., 2008) used on building sites, and a pole assignment implementation of the approach is utilized as the ‘baseline’ for the present research. In general, however, any inherent nonlinearities encountered in these examples of linear PIP control have been accounted for by ad hoc simulation-based adjustment of the weighting matrices. By contrast, for the present study, nonlinear SDP models are instead identified from experimental data and are represented using a novel time-varying ‘regulator’ NMSS form (Section 2). For the robotic platform considered in this paper, such models are shown to provide a good description of the nonlinear system dynamics but still yield low-order control algorithms of a similar implementational complexity to the existing PID and PIP approach. In fact, the ‘linear-like’ structure of the SDP-NMSS model means that it

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can be subsequently utilized to develop a time-varying (scheduled) SDP-PIP control algorithm. Although illustrated here for hydraulic manipulators, the approach is applicable to the general class of nth order SDP models with τ samples pure time delay. In this manner, the paper makes two new contributions: (i) generic results relating to the stability of the proposed nonlinear pole assignment algorithm for time delay SDP models (Section 3) and (ii) their application and evaluation for a hydraulically actuated nuclear decommissioning robotic manipulator (Section 4). It should be pointed out that an earlier paper considered SDP pole assignment in algebraic terms but was largely limited to simulation, together with brief consideration of the vibro-lance system mentioned above (Taylor, Chotai, & Young, 2009). By contrast, the present paper focuses on a dual-arm nuclear decommissioning robotic platform; utilises a slightly modified state space model; and, more significantly, expands from the brief results by Taylor, Chotai, and Burnham (2011) to derive the pole assignability and closed-loop stability of the algorithm in state space terms. This new approach depends on the introduction of a time-varying transformation matrix, as discussed in Section 3.

2. System representation This paper considers SDP models of the following nth order, discrete-time, difference equation form (Young, McKenna, & Bruun, 2001), yk ¼ a1 fχ k gyk1 …an fχ k gykn þ bfχ k gukτ

ð1Þ

where yk and uk are the output (joint angle) and input (applied voltage) variables, respectively, τ Z1 is the sampled time-delay, while bfχ k g and ai fχ k g ði ¼ 1; 2; …; nÞ are state-dependent parameters, assumed to be functions of a non-minimal state vector,

χ Tk ¼ ½u~ 1;k u~ 2;k …u~ r;k 

ð2Þ

in which u~ i;k ði ¼ 1…rÞ are available (measured) variables, usually (but not necessarily) derived from yk or uk. The form of the statedependency is problem specific but could, for example, be based on trigonometric or polynomial functions of the state variable(s). An illustration of its likely form is given in the following subsection. 2.1. System identification The focus of the present paper is on control system design, hence the system identification problem is only briefly summarized. For simple engineering devices, Eq. (1) can sometimes be derived from first principles. To illustrate, consider the following physically based, continuous-time model for a 1-link device (adapted from e.g. James, 1996, p. 749), MLx€ ¼ Mg sin xγ x_ þ FðtÞ

ð3Þ

where x is the link angle from the horizontal, x_ the angular velocity and F(t) an externally applied force. The coefficients g, L, M and γ represent gravitational acceleration, length, mass and a damping parameter, respectively. For SDP design, a suitable discrete-time model is obtained using a conventional forward difference approximation of Eq. (3), x1;k þ 1 ¼ x1;k þT s x2;k
 T sγ Tsg sin ðx1;k Þ þ uk x þ x2;k þ 1 ¼ 1 L ML 2;k

ð4Þ

where x1;k and x2;k are the sampled link angle and angular velocity, respectively; uk ¼ ðT s =MLÞF k is a control input variable, defined as a function of the external force Fk; and Ts is the sampling interval

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C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

(seconds). Straightforward algebra yields the following SDP model in the form of Eq. (1), with yk ¼ x1;k , n ¼ τ ¼ 2, b2 fχ k g ¼ b2 ¼ T s and, 

a1 χ k



Tsγ 2 ¼ a1 ¼ ML

  T s γ T 2s g sin ðyk2 Þ  a2 χ k ¼ 1 Lyk2 ML

ð5Þ

For the case of unity time delay τ ¼ 1, one straightforward representation of the system (1) is ð7Þ

where the n-dimensional (minimal) state vector is ð8Þ

For n ¼1, the transition matrix F1 fχ k g ¼ a1 fχ k g whilst for n 41,   afχ k g F1 fχ k g ¼ P in which afχ k g ¼ ½a1 fχ k g a2 fχ k g ⋯ an fχ k g and P ¼ ½In1 0 where In1 is an n1 identity matrix and 0 is a n1 column vector of zeros. Finally, g1 fχ k g ¼ ½bfχ k g 0 ⋯ 0T

where h ¼ ½1 0 ⋯ 0.

Time-delays τ 4 1 are introduced into the state space formulation by means of the following n þ τ1 dimensional (non-minimal) state vector: xk ¼ ½yk ⋯ ykn þ 1 uk1 ⋯ ukτ þ 1 T

ð10Þ

Here, xk þ 1 ¼ Fτ fχ k þ 1 gxk þ gτ uk

ð11Þ

with observation equation (9) and Fτ fχ k g partitioned as follows: Fτ fχ k g ¼ ½F2 fχ k gjF3 fχ k g where, F2 fχ k g ¼

"

F1 fχ k g

#

0τ1

with 0τ1 a matrix of zeros, and 2 3 0 ⋯ 0 bfχ k g 60 ⋯ 0 0 7 6 7 6 7 6⋮ ⋱ ⋮ ⋮ 7 6 7 6 7 0 7 F3 fχ k g ¼ 6 0 ⋯ 0 6 7 61 ⋯ 0 0 7 6 7 6 7 ⋮ 5 4⋮ ⋱ ⋮ 0 ⋯ 1 0 Note that F3 fχ k g has dimension ðn þ τ1Þ  ðτ1Þ and the ðn þ 1Þ th row of Fτ fχ k g consists entirely of zeros. Finally, the parameter vector associated with the input is time-invariant in this case, i.e., gτ ¼ ½0 ⋯ 0 1 0 ⋯ 0T where the (n þ 1)th row element of gτ is unity. 2.4. Control structure The state variable feedback control law associated with the above model takes the following form:

2.2. State space representation

xk ¼ ½yk yk1 ⋯ ykn þ 1 T

ð9Þ

2.3. Time-delays ð6Þ

Note that a2 fχ k g is not defined for yk -0. For this reason, the SDP control algorithm utilises a2 fχ k g ¼ 1T s γ =ðMLÞT 2s g=L for the limit, in a similar manner to Dutka, Ordys, and Grimble (2005). Taylor et al. (2009) consider SDP control of this physically based simulation model. The above analysis illustrates a mechanistic (or physically based) approach to obtaining the SDP model. The alternative is to use measured data collected from either planned experiments or from the normal operation of the plant. In this data-based approach to modelling, the SDP model is identified from experimental data using statistical methods, as demonstrated later in Section 4. The present paper utilises a previously developed SDP ‘back-fitting’ approach, as described by e.g. Young et al. (2001) and Taylor, Pedregal, Young, and Tych (2007) (see also the references therein). Here, the underlying model structure and potential state variables are first identified by statistical estimation of linear transfer function models from the experimental open-loop data. Such models take a similar form to Eq. (1) but with time invariant parameters. The second stage of this approach is based on the estimation of stochastic time-varying parameter models using conventional recursive Kalman Filtering and fixed interval smoothing, implemented within an iterative algorithm that involves re-ordering of the time series data in respect to each state variable (Taylor, Pedregal et al., 2007; Young et al., 2001). The non-parametrically defined nonlinearities are subsequently parameterized in some manner, in terms of their associated dependent variables. Finally, the coefficients of the parameterized model are estimated directly from input–output data using some method of dynamic model optimization. In Section 4, straightforward polynomial functions of the state variables are utilized, with the associated coefficients s estimated using fminsearch in Matlab .

xk þ 1 ¼ F1 fχ k þ 1 gxk þ g1 fχ k þ 1 guk

and the observation equation associated with (7) is yk ¼ hxk

uk ¼ cfχ k þ τ gxk þ wfχ k þ τ gr k

ð12Þ

where rk is the command input. For the purposes of the pole assignment algorithm developed later, the state variable feedback gain vector cfχ k þ τ g and open-loop command gain wfχ k þ τ g are expressed as functions of χ k þ τ where τ is the system time delay. The reason for this forward shift in the time index and its practical implications are discussed in Section 3. In Eq. (12), cfχ k þ τ g ¼ ½ffχ k þ τ gjgfχ k þ τ g

ð13Þ

where ffχ k þ τ g ¼ ½f 0 fχ k þ τ g f 1 fχ k þ τ g … f n1 fχ k þ τ g

ð14Þ

and gfχ k þ τ g ¼ ½g 1 fχ k þ τ g g 2 fχ k þ τ g … g τ1 fχ k þ τ g

ð15Þ

are vectors of control gains associated with the present and past values of the output and input variables, respectively. Since the functional representation of each parameter is estimated off-line, as discussed in Section 2.1, this approach yields a scheduled control system. One simplistic approach, for example, would be to utilize the well known discrete-time, algebraic Riccati equation at each sampling interval on an assumption of point-wise controllability (Taylor, Shaban et al., 2007). However, the optimality

C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

of this design is determined by the choice of model, with suboptimal solutions obtained in the general case. Furthermore, while some theoretical advances have been made regarding the asymptotic stability of the state-dependent Riccati equation approach in general, the conditions obtained can be difficult to fulfill (Banks, Lewis, & Tran, 2007). This limitation has motivated the development of the pole assignment approach below.

3. State-dependent pole assignment The present section proposes an algorithm for determining cfχ k þ τ g in Eq. (12), such that the closed-loop output response is stable and follows a predefined trajectory. Following an analogy with pole assignment design for linear (Young et al., 1987) and nonlinear (Taylor et al., 2009) non-minimal state space systems, albeit adapted here for this state-dependent regulator case, the new algorithm is written in the following general form:

Σfχ k þ τ g  ½ffχ k þ τ g; gfχ k þ τ gT ¼ βfχ k þ τ g

ð16Þ

where, gfχ k þ τ g ¼ ½g 1 fχ k þ τ g … g τ1 fχ k þ τ g

ð17Þ

with elements,   bfχ k þ τi g    gi χ kþτ gi χ k þ τ ¼ bfχ k þ τ g

ð18Þ

in which the coefficients g i fχ k þ τ g ði ¼ 1; …; τ1Þ are the control gains in Eq. (15). Note, for example, that with i ¼ τ1, the control gain given by Eq. (18) is defined as a function of both bfχ k þ τ g and bfχ k þ 1 g but, for notational simplicity, this gain is denoted g τ1 fχ k þ τ g. In other words, the time index represents the maximum forward shift of χ k þ τ that is required, in order to determine the value of g τ1 fχ k þ τ g, as required for implementation using Eq. (12). A similar convention is utilized for the other terms in this formulation. In Eq. (16), βfχ k þ τ g ¼ ½β1 fχ k þ τ g; …; β n þ τ1 fχ k þ τ gT is determined from the user selected design coefficients di,

βi fχ k þ τ g ¼ di ai fχ k þ τ g

ð19Þ

where ai fχ k þ τ g ¼ 0 for i 4 n. Propositions 2 and 3 below will show that di are associated with the eigenvalues of the desired time invariant (linear) closed-loop system. Finally, Σfχ k þ τ g is a matrix of dimension ðn þ τ1Þ  ðn þ τ1Þ, defined as follows:

Σfχ k þ τ g ¼ ½Σb fχ k þ τ g; Σa fχ k þ τ1 g where,

"

0τ1;n

ð20Þ

#

Σb fχ k þ τ g ¼ I bfχ g n kþτ in which 0τ1;n is matrix of zeros and In an identity matrix, while the ðn þ τ1Þ  τ1 dimensional Σa fχ k þ τ1 g is defined as 2 3 1 0 ⋯ 0 6 a 1 fχ 7 1 ⋯ 0 k þ τ 1 g 6 7 6 7 6 a2 fχ k þ τ1 g 7 a1 fχ k þ τ2 g ⋯ 0 6 7 6 7 ⋮ ⋮ ⋮ 6 7 6 7 6 an fχ k þ τ1 g an1 fχ k þ τ2 g ⋯ a 2 fχ k þ 1 g 7 6 7 6 ⋯ a 3 fχ k þ 1 g 7 0 an fχ k þ τ2 g 6 7 6 7 6 7 ⋮ ⋮ ⋮ 6 7 6 0 0 ⋯ an1 fχ k þ 1 g 7 4 5 0 0 ⋯ a n fχ k þ 1 g Propositions 2 and 3 below show that the forward shift of the parameters in Eqs. (16)–(20) is necessary to achieve the desired response. In this regard, it is important to recall their state

1719

dependent form. For many engineering devices, notably the robotic manipulators considered in this paper, these parameters are functions of the delayed input and output signals, hence a forward shift does not cause problems in practice. To illustrate using the SDP model for a 1-link device mentioned above, τ ¼ 2 while a2 fχ k þ 2 g in Eq. (6) is a function of the measured joint angle yk. To ensure unity steady state gain in the ideal case, the openloop command gain is determined as follows:   1 þd1 þ ⋯ þdn þ τ1 w χ kþτ ¼ bfχ k þ τ g

ð21Þ

In general, Type 1 servomechanism behavior can be achieved by the introduction of an integral-of-error state variable (Young et al., 1987) into either (7) or (11). However, such a state variable is omitted from the present formulation since it is not necessary for the application considered below. In fact, the deadband calibration routine ensures that there is no movement when uk ¼ 0, whilst the destabilizing nature of integral action potentially has a negative impact on the control performance. For non-singular Σfχ k þ τ g (see Proposition 1 below), Eqs. (16) and (18) can be numerically solved at each sampling instant to determine the state variable feedback control gains ffχ k þ τ g and gfχ k þ τ g, for implementation using Eq. (12). Alternatively, it is straightforward to derive an algebraic solution (off-line) and then implement this solution on-line at each sampling instant, so reducing the computational burden in practical applications. This aspect of the approach is demonstrated in Section 4. Proposition 1: pole assignability The pole assignment algorithm defined by Eqs. (16) and (18) can be solved if and only if bfχ k þ τ g a0 for all k. Demonstration In order to solve Eqs. (16) and (18), it is obvious that the two conditions are: bfχ k þ τ g a 0 and Σfχ k þ τ g is non-singular. In fact, examination of Σfχ k þ τ g in Eq. (20) shows that the two conditions are equivalent. Alternatively, successive substitutions in the manner of Kuo (1980), using the open-loop state equation (11), yields the following controllability matrix for τ 4 1 (a similar matrix is defined for τ ¼ 1 but is omitted for brevity): Sfχ k þ τ g ¼ ½gτ ; Fτ fχ k þ τ ggτ ; Fτ fχ k þ τ gFτ fχ k þ τ1 ggτ ; ⋯; Fτ fχ k þ τ g⋯Ffχ k þ 2 gFfχ k þ 1 ggτ 

ð22Þ

Straightforward algebraic manipulation shows that Sfχ k þ τ g is nonsingular if and only if bfχ k þ τ g a 0. Noting Eq. (1), these results are equivalent to the trivial requirement that the control input variable has some influence on the system output. Proposition 2: closed-loop for τ ¼ 1 Assuming no model mismatch and τ ¼ 1, the control algorithm (12), with cfχ k þ 1 g obtained using Eqs. (16) and (18), applied to the nonlinear model (1) or its equivalent (7), yields an output response defined by the following linear, discrete-time difference equation: ~ yk ¼ d1 yk1 ⋯dn ykn þ dr k1

ð23Þ

where d~ ¼ 1 þ d1 þ ⋯ þ dn and di are the user selected design coefficients. Expressed as a linear transfer function, Eq. (23) has poles pi ði ¼ 1⋯nÞ. With stable design poles, closed-loop stability clearly follows.

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C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

and

Demonstration Substituting Eq. (12) into to the open-loop NMSS model (7), yields the closed-loop control system, xk þ 1 ¼ Axk þ wfχ k þ 1 gg1 fχ k þ 1 gr k ;

yk ¼ hxk

ð24Þ

where A ¼ F1 fχ k þ 1 gg1 fχ k þ 1 gcfχ k þ 1 g is a n  n time invariant (linear) state transition matrix with the following form: 2 3 d1 d2 ⋯ dn1 dn 6 1 0 ⋯ 0 0 7 6 7 6 7 6 0 1 ⋯ 0 0 7 6 7 A¼6 ð25Þ ⋮ ⋯ ⋮ ⋮ 7 6 ⋮ 7 6 7 4 0 0 ⋯ 0 0 5 0

0

⋯

1

0

T ~ . Since the eigenvalues of A Note that wfχ k þ 1 gg1 fχ k þ 1 g ¼ ½d0⋯0 are equivalent to pi ði ¼ 1⋯nÞ, the closed-loop response is given by Eq. (23) as required. This trivial result for the τ ¼ 1 case has a direct analogy with conventional linear pole assignment, i.e. the control gains may be obtained by straightforwardly equating F1 fχ k þ 1 gg1 fχ k þ 1 gcfχ k þ 1 g with the desired closed-loop transition matrix A.

Proposition 3: closed-loop for τ 4 1 Assuming no model mismatch and τ 4 1, the control algorithm (12), with cfχ k þ τ g obtained using Eqs. (16) and (18), applied to the nonlinear model (1) or its equivalent (11), yields the desired output response, ~ yk ¼ d1 yk1 ⋯dn þ τ1 yknτ þ 1 þ dr kτ

ð26Þ

where d~ ¼ 1 þ d1 þ ⋯ þ dn þ τ1 . With stable design poles pi ði ¼ 1⋯n þ τ1Þ, closed-loop stability follows. Demonstration Substituting Eq. (12) into to the open-loop NMSS model (11), yields the closed-loop control system, xk þ 1 ¼ Afχ k þ τ gxk þ wfχ k þ τ ggτ r k ;

yk ¼ hxk

ð27Þ

where the transition matrix Afχ k þ τ g ¼ Fτ fχ k þ 1 ggτ cfχ k þ τ g is time-varying (state-dependent). Standard concepts in linear control theory relating to the closed-loop eigenvalues and stability do not apply. For example, it is well known that, for nonlinear systems, the eigenvalues do not necessarily determine the stability and performance of the closed-loop system. However, using the special control formulation proposed above, it is straightforward to check that Afχ k þ τ g can always be decomposed into the following form: Afχ k þ τ g ¼ Tfχ k þ τ gDT1 fχ k þ τ1 g

ð28Þ

Note that the closed-loop transition matrix is composed of Tfχ k þ τ g and Tfχ k þ τ1 g but, following a similar approach to that noted above, is denoted simply Afχ k þ τ g. In Eq. (28), D ¼ ½D1 D2 , in which, 2 3 0 0 ⋯ 0 6 1 7 0 ⋯ 0 6 7 6 7 6 0 7 1 ⋯ 0 6 7 6 ⋮ 7 ⋮ ⋯ ⋮ 6 7 6 7 d ⋯ d d D1 ¼ 6 ð29Þ τ τþ1 τ þ n1 7 6 7 6 7 0 ⋯ 0 6 0 7 6 7 6 ⋮ 7 ⋮ ⋯ ⋮ 6 7 6 7 0 ⋯ 0 4 0 5 0 0 ⋯ 0

2

0 6 0 6 6 6 0 6 6 ⋮ 6 6 D2 ¼ 6 6 d1 6 6 1 6 6 ⋮ 6 6 4 0 0

⋯

0

0

⋯

0

0

⋯

0

0

⋱ ⋯

⋮ dτ3

⋮ dτ2

⋯

0

0

⋱

⋮

⋮

⋯

1

0

⋯

0

1

1

3

7 7 7 0 7 7 ⋮ 7 7 7 dτ1 7 7 7 0 7 7 ⋮ 7 7 7 0 5 0 0

ð30Þ

is a time-invariant matrix with eigenvalues pi. Here, the matrix Tfχ k þ τ g ¼ ½T1 fχ k þ τ gjT2 fχ k þ τ g, where 2 3 1 0 0 ⋯ 0 6 7 0 1 0 ⋯ 0 6 7 6 7 6 7 ⋮ ⋮ ⋮ ⋮ 6 7 6 7 0 0 0 ⋯ 1 6 7 6 a i fχ 7 ai þ 1 fχ k þ τ g k þτg 6 7   6 0 ⋯ 0 7 χ g χ g bf bf T1 χ k þ τ ¼ 6 7 kþτ kþτ 6 7 6 7 ⋮ ⋮ ⋮ ⋮ 6 7 6 a 2 fχ k þ 3 g 7 a3 fχ k þ 3 g a4 fχ k þ 3 g 6 7 ⋯ 0 6 bfχ k þ 3 g 7 χ g χ g bf bf kþ3 kþ3 6 7 6 a fχ a2 fχ k þ 2 g a3 fχ k þ 2 g a n fχ k þ 2 g 7 4 1 k þ 2g 5 ⋯ bfχ k þ 2 g bfχ k þ 2 g bfχ k þ 2 g bfχ k þ 2 g in which ai fχ k g ¼ 0 for i 4 minðτ1; nÞ and 2 0 0 ⋯ 6 0 0 ⋯ 6 6 6 ⋮ ⋮ 6 6 0 0 ⋯ 6 6 a1 fχ k þ τ g 1 6   6 ⋯ T2 χ k þ τ ¼ 6 bfχ k þ τ g bfχ k þ τ g 6 6 ⋮ ⋮ 6 6 1 6 0 0 6 bf χ k þ 3g 6 6 4 0 0 0

0

3

7 7 7 7 ⋮ 7 7 0 7 a i fχ k þ τ g 7 7 bfχ k þ τ g 7 7 7 7 ⋮ 7 a 1 fχ k þ 3 g 7 7 bfχ k þ 3 g 7 7 7 1 5 bfχ k þ 2 g 0

in which ai fχ k g ¼ 0 for i 4 n. Hence, pre  multiplying the state equations (27) by T1 fχ k þ τ g and rearranging yields x~ k þ 1 ¼ Dx~ k þ wfχ k þ τ gT1 fχ k þ τ ggr k

ð31Þ

where x~ k ¼ T fχ k þ τ1 gxk . The closed  loop transition matrix is now in a linear time invariant form D, albeit for a transformed ~ hence state vector x~ k . Note that wfχ k þ τ gT1 fχ k þ τ gg ¼ dg, 1

~ x~ knτ þ 2 ¼ Dx~ knτ þ 1 þ dgr knτ þ 1

ð32Þ

Successive substitutions using (32) yields ~ ~ x~ k þ 1 ¼ Dn þ τ x~ knτ þ 2 þ Dn þ τ1 dgr knτ þ 2 þ ⋯ þ dgr k Note from the characteristic polynomial of D and the Cayley Hamilton theorem that Dn þ τ1 þ d1 Dn þ τ2 þ ⋯ þ dn þ τ2 D þ dn þ τ1 I ¼ 0 i.e. a matrix of zeros. Hence, taking x~ k þ d1 x~ k1 þ ⋯ þdn þ τ1 x~ knτ þ 1 and re arranging yields x~ k ¼ d1 x~ k1 d2 x~ k2 ⋯dn þ τ1 x~ knτ þ 1 ~ þðDn þ τ2 þd Dn þ τ3 þ ⋯ þ d IÞdgr 1

n þ τ2

~ ~ þ⋯ þ ðDþ d1 IÞdgr k1 þ dgr k

knτ

ð33Þ

Note from the observation equation, yk ¼ hxk ¼ hT

1

~ χ kg fχ k gx~ ¼ hxf

ð34Þ

C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

that the transformation does not affect the first element of the state vector. Hence, it is a trivial matter to obtain the output response from Eqs. (33) and (34). In this regard, hðDn þ τ2 þ d1 Dn þ τ3 þ ⋯ þ ~ ¼ 0, hdg ~ ¼ 0 (etc.) while hðDτ1 þ d Dτ2 þ ⋯ þ d IÞ dn þ τ2 IÞdg 1 τ1 ~ ¼ d~ and Eq. (34) reduces to Eq. (26) as required. dg

4. Experimental results The experimental results are based on a dual-arm mobile nuclear decommissioning robot. The Brokk-40 base machine is 650 mm wide, allowing for access through narrow doorways, and consists of a moving vehicle, hydraulic tank, remote control system and (initially) single manipulator. The unit is electrically powered to facilitate internal use, with an onboard hydraulic pump to power the caterpillar tracks and, by means of several hydraulic pistons, the manipulator. However, for nuclear decommissioning tasks, accessing the robot on-site to change tools can be a slow, laborious and potentially hazardous task. A more flexible system with the ability to perform multiple tasks without a tool change has been achieved with the mounting of two HydroLek-7W manipulators. Each manipulator has 6 degrees-of-freedom with a continuous (3601) jaw rotation mechanism, together with a dual function gripper fitted with a pressure sensor. The end-effectors can be equipped with a variety of tools, such as percussive breakers, hydraulic crushing jaws, excavating buckets and concrete milling heads. Without tools, each arm has a weight of 45 kg and the lift capacity at full reach is 150 kg. The azimuth yaw, shoulder pitch, elbow pitch, forearm roll and wrist pitch joints, illustrated in Fig. 1, are all fitted with potentiometer feedback sensors, allowing the position of the end-effector to be determined during operation. The joints are actuated via hydraulic pistons, which are powered via an auxiliary output from the hydraulic pump of the base unit. 4.1. Software interface A standard input device, such as a joystick, is connected to a PC running a graphical user interface developed for the National Instruments (NI) Labview software environment. The PC transmits information to a NI Compact Fieldpoint Real-Time controller (cFP)

1721

via an Ethernet networking connection. The cFP is a stand alone device running a real-time operating system, allowing for the precise sampling rates needed for discrete-time control. The hydraulic pistons are controlled by seven pairs of control valves, where each pair has an input for both positive and negative flow. Output modules convert the digital cFP signal to a varied voltage fed to the control valves. The present authors have developed a semi-automated system for calibrating and initializing the robot for open-loop data collection (Taylor et al., 2010). Here, the robot is manipulated into a suitable configuration using standard proportional controllers. The operator selects from classical step experiments or pseudorandom signals for the estimation of nonlinear models. For each joint, the system input uk is scaled to lie in the range  100, representing the maximum power in a negative ‘closing’ direction, through to þ100, representing the maximum power in a positive ‘opening’ direction. The dead-band of the system is eliminated by the input calibration step, hence an input of zero represents no movement. Finally, the output yk is the potentiometer voltage, representing a scaled joint angle. 4.2. Kinematics Since it has no bearing on the end-effector location, the gripper is neglected from the kinematic solution illustrated in Fig. 2. Hence, each manipulator is described as a kinematic model with six solid links and rotational joints. In fact, the manipulator is overspecified in terms of end-effector positioning, with the additional degrees-of-freedom potentially utilized to allow the end-effectors orientation to be set and to reach past obstacles. However, to demonstrate the practical feasibility of the proposed SDP algorithm, consider a reduced 2-DoF system that allows for straightforward movement of the end-effector in a plane. Fig. 3 shows how the manipulator is limited to a 2-DoF system for Joints 2 and 3, with the remaining joints locked off. In this case, the forward kinematic relationship of the joint angles to the horizontal Px and vertical Pz positions of the end-effector is given by P x ¼ Rc23r þ a2 c2 þ a1

ð35Þ

P z ¼ Rs23r þ a2 s2

ð36Þ

where c2 and s2 denote cos ðθ2 Þ and sin ðθ2 Þ, c23r represents cos ðθ2 þ θ3 θR Þ, θR ¼ a tan 2ða3 þa4 a5 ; d4 þd5 Þ and R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða3 þ a4 a5 Þ2 þ ðd4 þ d5 Þ2 , in which θ2 and θ3 represent the shoulder and elbow pitch, respectively (Fig. 1), while ai ði ¼ 1; …; 5Þ and ðd4 ; d5 Þ are link lengths (Fig. 3). The inverse kinematics are a1

J1

a2

J2

a3

J3

J4 a4

d4

J5 a5 d5 J6

Fig. 1. HydroLek-7W rotational joints.

Fig. 2. 6-DOF kinematic description of HydroLek-7W manipulator.

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C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

a1

a2

J2

a3+a4-a5

J3

800

θR

600

400 R

d4+d5

Pz (mm)

200

0

−200 Fig. 3. Simplified 2-DOF description of HydroLek-7W manipulator for illustrative resolved motion experiment.

−400 straightforwardly derived as follows: c3 ¼

ðP x a1 Þ2 þ P 2z R2 a22 2a2 R qffiffiffiffiffiffiffiffiffiffiffi 1c23

ð38Þ

θ3 ¼ a tan 2ðs3 ; c3 Þ þ θR

ð39Þ

s3 ¼ 7

c2 ¼

ðP x a1 ÞðRc3 þ a2 Þ þ P z Rs3 d

ð40Þ

s2 ¼

P z ðRc3 þ a2 ÞðP x a1 ÞRs3 d

ð41Þ ð42Þ

4.3. System identification Preliminary open-loop step experiments suggest that a first order linear difference equation, i.e.,

600 800 Px (mm)

1000

1200

Joint

p1

p2

p3

θ2 θ3

 2.2898e  007  1.9290e  007

1.2576e  005 1.8214e  005

0.0055 0.0051

3 X 10ˉ

6 4 2 ˉ100

ˉ 50

ð43Þ

provides an approximate representation of individual joints, with time-invariant parameters fa1 ; bg and the time delay τ depending on the sampling interval Δt. However, it is readily apparent that the numerical values of fa1 ; bg are not repeatable for different step experiments. Hence, various candidate SDP structures have been investigated for open-loop movement of the right hand side manipulator shoulder and elbow pitch, moving in air with no additional loading terms. Trial and error experimentation suggests that a sampling interval Δt ¼ 0:07 s yields a satisfactory compromise between a fast response and the following low-order model for control system design: yk ¼ a1 fχ k gyk1 þbfχ k guk2

400

Table 1 Optimized polynomial coefficients.

b

where d ¼ ðRc3 þ a2 Þ2 þ ðRs3 Þ2 . Utilizing these kinematic equations and the practical limits for each joint, Fig. 4 shows the end-effector workspace and the figure-of-eight positional commands for a resolved motion experiment discussed later. In Fig. 4, the position ð0; 0Þ denotes the base of link 1, i.e. where Joint 1 in Fig. 2 attaches to the backplate connecting the HydroLek manipulator to the Brokk base unit.

yk ¼ a1 yk1 þ bukτ

−800 200

Fig. 4. HydroLek-7W manipulator workspace (thin trace) and illustrative resolved motion trajectory (thick), plotted on 2-dimensional plane.

Similarly,

θ2 ¼ a tan 2ðs2 ; c2 Þ

−600

ð37Þ

ð44Þ

The model is based on Eq. (1) with n¼ 1, τ ¼ 2 and initially χ k ¼ ½yk1 uk3 . In fact, the statistical estimates suggest that a1 fχ k g is relatively constant over time, with the pole close to unity.

0 input

50

100

Fig. 5. SDP relationships for Joint 2 (dashed) and Joint 3 (solid) showing the parameter value plotted against input voltage (scaled 100- þ 100).

Assuming a1 ¼ 1, b2 fχ k g represents the state-dependent angular velocity approximated by bfχ k g ¼ p1 u2k3 þ p2 uk3 þ p3

ð45Þ

The time-invariant coefficients pi are optimized from the experis mental data using fminsearch in Matlab , yielding the estimates shown in Table 1 and Fig. 5. The model responses (43) and (44) are compared with the measured data for an illustrative open-loop shoulder joint experiment in Fig. 6. The SDP model is far superior, typically explaining over 95% of the output variance, compared with only 20–60% for the linear model.

C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

algorithm is

4.4. Control system design

    bfχ k þ 1 g uk ¼ f 0 χ k þ 2 yk  g u þ w χ k þ 2 rk bfχ k þ 2 g 1 k1

The SDP-NMSS (11) representation of Eq. (44) is " xk ¼

#

yk

 ¼

uk1

a1

bfχ k g

0

0

   0 xk1 þ uk1 1

ð46Þ

and yk ¼ ½1 0xk . For bfχ k g 4 0, Eq. (16) becomes "

f 0 fχ k þ 2 g

#

" ¼

g1

1723

0

1

bfχ k þ 2 g

a1

#1 "

p1 p2 a1

# ð47Þ

p1 p2

in which ðzp1 Þðzp2 Þ ¼ z2 ðp1 þ p2 Þz þ p1 p2 is the desired closedloop characteristic polynomial (26), with the poles fp1 ; p2 g chosen to lie within the unit circle of the complex z-plane. Utilizing Eqs. (12), (18) and (21), together with a1 ¼ 1, the SDP control

ð48Þ

where f 0 fχ k þ 2 g ¼ wfχ k þ 2 g ¼ ðp1 p2 g 1 Þ=bfχ k þ 2 g and g 1 ¼ p1  p2 a1 . Note that g 1 is time-invariant in this case, whilst the state dependent parameter bfχ k þ 2 g is determined on-line using Eq. (45). Finally, three linear control systems are also considered. Experimental results suggest that the most robust linear algorithm is based on an operating level of uk ¼ 0, hence utilizing Eq. (45) and Table 1, b2 ¼0.0055 and b2 ¼ 0.0051 for the shoulder and elbow joints, respectively. For comparison, linear controllers are also developed for the maximum input signal each direction, i.e. Eq. (45) with uk ¼ 100 and uk ¼100. Eq. (47) is solved off-line to determine the time-invariant coefficients of the linear controllers. 4.5. Closed-loop results

100 input

50 0 ˉ 50 ˉ100

0

10

20

0

10

20

30

40

50

60

30 40 time (seconds)

50

60

output

15 10 5

Fig. 6. Upper subplot: input voltage (scaled 100- þ 100) plotted against time. Lower subplot: potentiometer voltage (V), showing the optimized SDP model response (thick trace), experimental data (thin) and linear model (dashed).

Typical practical results for the SDP-NMSS controller are illustrated by Figs. 7 and 8, showing a satisfactory response across a range of operating levels. For these illustrative results, the design poles p1 ¼ 0.9 and p2 ¼0.94 are chosen by trial and error at the simulation stage. In fact, these poles are useful for comparative purposes, since they correspond to the fastest response achievable by the existing (linear) controllers for this manipulator without activating the input saturation levels. Linear controllers based on operating levels of uk ¼ 0 or 100 generally yield slower responses than the SDP algorithm, since the associated linear model tends to ‘overestimate’ the actual, time-varying, steady state gain of the system (see Fig. 5). Similarly, the linear controller based on uk ¼ 100 works adequately for fast movements in a negative direction but tends to overshoot the set point at other times, as illustrated by Fig. 8. It should be emphasized that the control objective is to follow the user-defined response given by the design poles. This is shown by the dashed trace in Fig. 8, obtained in this case by simulating a second order transfer function with poles p1 ¼0.9 and p2 ¼0.94 in

6

x 10ˉ3

15

10

b2

output

5.5

4.5

5 0

50

100

4

150

100

0

50

100

150

0

50 100 time (seconds)

150

1.4

50

1.3

0

f0

input

5

ˉ 50

1.2 1.1

ˉ100

0

50 100 time (seconds)

150

1

Fig. 7. Clockwise from upper left: closed-loop experiment (i.e. using the robot demonstrator) showing the response (potentiometer voltage, V) of joint angle (thick trace) to a sequence of step changes in the command (thin), b2 fχ k g, f 0 fχ k g and input voltage uk (scaled 100- þ 100), all plotted against time.

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C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

output

15

200

10 5

100 0

10

20

30

0

10

20

100

0

input

50 0

−100

ˉ100

0

10

20

30 0 time (seconds)

10

20

Fig. 8. Closed-loop response of joint angle to positive (joint opening) and negative (closing) step changes in the command input, showing experimental data (i.e. from the robot demonstrator) obtained using the linear (thin trace) and nonlinear (thick) controllers, compared to the simulated design response (dashed), all plotted against time. Upper subplot: potentiometer voltage (V). Lower subplot: input voltage (scaled 100- þ 100).

Pz (mm)

ˉ 50

−200

−300

−400 Table 2 Mean absolute error between joint angle and ideal response, comparing the new SDP approach with the baseline linear algorithm (uk ¼0) and linear algorithms optimized for two other operating levels (uk ¼ 100 and uk ¼100). Experiment

SDP

uk ¼ 100

uk ¼ 0

uk ¼100

Fig. 7 Fig. 8

0.1640 0.0921

0.5229 0.2289

0.2414 0.1030

0.2400 0.1346

−500

−600 600

open-loop. Although arbitrarily chosen for the present paper, which focuses on the low-level joint control problem, the poles and command sequences are generally determined by the higherlevel control module for optimizing tasks using the attached tools. Hence, the faster speed of response and overshoot for the linear controller seen in Fig. 8, represents an undesirable deviation from the required behavior of the system. Table 2 summarizes this error between the experimentally observed joint angle and the ideal response for a set of experiments similar to Figs. 7 and 8, demonstrating the improved performance of the SDP-NMSS design against all three linear controllers. Finally, Fig. 9 illustrates a straightforward resolved motion experiment, in which the end-effector is programmed to trace a circle in a clockwise motion, followed by a second lower circle in an anticlockwise direction. The end-effector commands are based on point-to-point motion, with the trajectory defined by a series of incremental steps in the positional set point for ½px ; pz . The speed of response is determined by the number of iterations and the waiting time for each point. This straightforward approach to positional trajectory planning appears to work well in practice for this system. Once the experimental data are displayed in Cartesian form on the work-plane, as in Fig. 9, the potential benefits of nonlinear control become more apparent, particularly when relative fast movement is required. For the illustrative results shown, each circle was programmed to be completed in 3 s, with the experiment repeated three times in Fig. 9.

5. Conclusions This paper has developed a pole assignment algorithm for the control of nonlinear dynamic systems described by State Dependent Parameter (SDP) models, with a particular focus on the hydraulically actuated joints of a mobile robot designed for

700

800 Px (mm)

900

1000

Fig. 9. End-effector location associated with Fig. 4, showing experimental data (i.e. from the robot demonstrator) obtained using the linear (dashed) and nonlinear (solid) controllers, together with the set point generated by the inverse kinematics (gray).

nuclear decommissioning. The new approach can be applied to nth order SDP models with τ Z 1 sampled time-delays. However, analysis of experimental data suggests that a first order SDP model with state-dependent gain, representing the angular velocity, is adequate for fast and smooth control of the manipulator joints. Integral action is not required for the hydraulic systems considered in this paper, hence the SDP model is represented in a regulator, discrete-time, non-minimal state space (NMSS) form, in which the time-delays are handled automatically. Using the proposed pole assignment approach, the paper shows that the closed-loop system reduces to a linear transfer function with the specified poles. Hence, global stability of the nonlinear system is guaranteed at the design stage. The time indices of the state dependent parameters, which are shifted forward by τ samples in the control formulation, are the key to obtaining an appropriate closed-loop response. Algebraic solutions can be derived off-line to yield a practically useful control algorithm that is relatively straightforward to implement on a digital computer, requiring only the storage of delayed system variables, coupled with straightforward arithmetic expressions in the control software. Links with feedback linearization (Kreutz, 1989), state dependent Riccati equation (Banks et al., 2007) and other nonlinear and adaptive approaches (e.g. Mohanty & Yao, 2011a, 2011b) are being investigated by the authors. Of particular interest are their relative performance and robustness when applied to the robotic manipulator, and these results will be reported in future papers. In this

C.J. Taylor, D. Robertson / Control Engineering Practice 21 (2013) 1716–1725

regard, the authors are also investigating other manipulator configurations and settings (e.g. various loads) with a view to identify additional state variables and potential multivariable state-dependencies. Nonetheless, to illustrate the modeling approach and typical closed-loop results, experimental data associated with resolved motion for the shoulder and elbow pitch of one manipulator have been considered in the present paper. The authors are now investigating the remaining joints, with a view to the development of a high-level control algorithm for resolved control of the dual-manipulators. In the future, these algorithms are intended to be part of a self-calibrating automatic control system suitable for a range of activities relating to nuclear decommissioning. Acknowledgments This work was supported in part by the UK Nuclear Decommissioning Authority (NDA). The authors are also grateful for the support of Brokk UK Ltd. and Hydro-Lek Ltd. The statistical tools have been assembled as the CAPTAIN Toolbox (Taylor, Pedregal s et al., 2007) within the Matlab software environment and may be obtained from the first author. References Alleyne, A., & Liu, R. (2000). A simplified approach to force control for electrohydraulic systems. Control Engineering Practice, 8, 1347–1356. Bakari, M., Zeid, K., & Seward, D. (2007). Development of a multi-arm mobile robot for nuclear decommissioning tasks. International Journal of Advanced Robotic Systems, 4, 387–406. Bakari, M. J. (2008). Development of a multi-arm mobile robot for nuclear decommissioning tasks. Ph.D. Thesis, Engineering Department, Lancaster University, UK. Banks, H. T., Lewis, B. M., & Tran, H. T. (2007). Nonlinear feedback controllers and compensators: A state-dependent Riccati equation approach. Computational Optimization and Applications, 37, 177–218. Bock, T., Stricker, D., Fliedner, J., & Huynh, T. (1996). Automatic generation of the controlling-system for a wall construction robot. Automation in Construction, 5, 15–21. Bradley, D. A., & Seward, D. W. (1998). The development, control and operation of an autonomous robotic excavator. Journal of Intelligent Robotic Systems, 21, 73–97. Budny, E., Chlosta, M., & Gutkowski, W. (2003). Load-independent control of a hydraulic excavator. Journal of Automation in Construction, 12, 245–254. Dutka, A. S., Ordys, A. W., & Grimble, M. J. (8–10 June 2005). Optimized discretetime state dependent Riccati equation regulator. In American control conference (pp. 2293–2297). USA: Portland, OR. Feng, Y., Yu, X., & Man, Z. (2002). Non-singular terminal sliding mode control of rigid manipulators. Automatica, 38, 2159–2167. Gonzalez, A. H., Perez, J. M., & Odloak, D. (2009). Infinite horizon MPC with nonminimal state space feedback. Journal of Process Control, 19, 473–481. Gu, J., Taylor, C. J., & Seward, D. W. (2004). The automation of bucket position for the intelligent excavator LUCIE using the Proportional-Integral-Plus (PIP) control strategy. Journal of Computer-Aided Civil and Infrastructure Engineering, 12, 16–27. Guan, C., & Pan, S. (2008). Nonlinear adaptive robust control of single-rod electrohydraulic actuator with unknown nonlinear parameters. IEEE Transaction on Control Systems Technology, 16, 434–445. Ha, Q., Nguyen, Q., Rye, D., & Durrant-Whyte, H. (2000). Impedance control of a hydraulic actuated robotic excavator. Journal of Automation in Construction, 9, 421–435. Hashemi, S., Abbas, H., & Werner, H. (2012). Low-complexity linear parametervarying modeling and control of a robotic manipulator. Control Engineering Practice, 12, 248–257. Iborra, A., Pastor, J. A., Alvarez, B., Fernandez, C., & Merono, J. M. F. (2003). Robots in radioactive environments. IEEE Robotics and Automation Magazine, 10, 12–22.

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