A task-level approach for the position control of macro-mini robotic systems

Abstract

This article proposes a new task level approach to control the position of macro-mini robotic systems. The proposed method combines redundancy resolution and PI compensation. Taking advantage of the redundancy resolution, this control method accounts for the differences in the dynamics of the macro robot and the mini robot during positioning tasks while ensuring that both robots respect their mechanical limits. The combination with a PI compensation improves the control performance in tracking tasks. This work also presents the adaptation of two other separate methods, namely mid-ranging control and Model Predictive Control, to macro-mini robots. A new macro-mini robot is introduced and experiments, in which the three control methods perform the same trajectories, are conducted. The control performance as well as the advantages or drawbacks of each method are compared. The general behavior of each robot component is also observed and analyzed. Finally, several practical issues and potential future work are discussed.

Introduction

Many applications in robotics require robots to have a large workspace, especially in industry. In many circumstances, the existence of singularities is the main challenge to enlarge the workspace of robots, especially for parallel robots. Redundancy is a potential strategy that has shown many promising results in avoiding singularities and in extending both translational and rotational workspaces of serial robots (see for example Ref. ¹ ), and of parallel robots such as Refs.^2,3 On the other hand, the workspace’s volume is proportional to the dimension of a robot. Larger robots deal with higher inertia which makes it much more difficult to increase the operational speed. The solution is not as simple as adding more powerful actuators because this problem requires working closely on both sides: the mechanical design and the control system. Once this collaboration is successfully implemented, the speed is significantly enhanced, particularly in studies involving parallel robots such as Refs.^4–7 Nonetheless, there is no perfect robot that can have both dominant features: high operational speed and a large workspace. The idea of a macro-mini manipulator was proposed in this context. A macro-mini manipulator is technically a special case of redundancy, which consists of two robots assembled in a serial configuration. It combines two robots that have very different and complementary properties. Then, the output of the macro-mini system can earn the advantages of each robot, as well as reduce their weaknesses. As suggested by the name, the macro is bigger and attached to the base. On the contrary, the mini is much smaller. Thus, fast and dexterous robots are usually selected.

The concept of macro-mini robot architecture was initially introduced in Ref., ⁸ which was aiming to enhance the accuracy of position control. Several recent studies considering macro-mini systems working in the range of micrometers also exploit this advantage, such as Refs.^9,10 In Ref., ¹¹ the authors proved that using this architecture can significantly increase the bandwidth of position control and force control. It also enhances the interaction capability. Later, in Refs.,^12,13 impedance control for a macro-mini system was presented. Robotic polishing or milling are popular applications that require accurate force control. Therefore, the macro-mini architecture can be applied, for example in Ref. ¹⁴ During the decade following its inception, the idea of macro-mini was bonded with the combination of two serial rigid robots. However, this architecture quickly became diverse afterward with different compositions, as long as the principle mentioned above is applicable. For example, in Ref., ¹⁵ the combination of a macro flexible robot and a mini rigid robot was presented, in Refs.,^16,17 that of two parallel robots was proposed (one of them can be a cable-driven robot). The parallel configuration of the macro-mini system was also proposed, where the two robots can be mounted on a common base and work on the same task, like one of the two configurations introduced in Ref. ¹⁸ This configuration enables to considerably extend the force bandwidth of the combined system as demonstrated in Ref. ¹⁹ In Refs.,^20,21 the authors proposed the use of a macro robot with a passive mini parallel mechanism for physical human-robot interaction (pHRI). This architecture has clearly demonstrated promising results in this domain.

From a mechanical standpoint, the accurate control of the position of a robot with a high bandwidth requires agility. However, for tasks that require a large workspace, the robot needs to be large and therefore heavy, which makes it difficult to produce motion with a high bandwidth. This is the main reason for taking a macro-mini approach in this work. Only the lightweight mini robot needs to be able to operate at a high bandwidth, which is much more feasible. This approach has a significant impact on the controller, namely the proposed PI + RR paradigm can be used which in essence implements a synergy between mechanical design and control, based on the dynamic modeling of the system.

The dynamic model plays an important role in most control methods, from classical to modern, as discussed in Ref. ²² Model-based approaches certainly yield great results but the analytical dynamic model may not be available. Recently, data-driven modeling like identification techniques is preferred and the dynamic model can be computed numerically. For that reason, in this study, the macro-mini robot is revisited and the motion control problem is considered systematically to overcome existing drawbacks. Firstly, the position control of each robot is assumed to be given or solved separately. The subsystem that includes the macro or the mini and their position controller is then modeled as a linear system. In order to take dynamic effects into account, the model of each subsystem is identified while considering its interaction with the other. After this stage, the combined system is seen as a mathematical model. The control techniques that work on top of this model are investigated. In addition to simulations, the control scheme of a new macro-mini system is introduced. Experiments are carried out on this system to verify control performance as well as to highlight practical implementation issues for each method.

The rest of this paper is structured as follows. Section 2 first describes the kinematics of the macro-mini system. A mathematical model is then derived based on several assumptions and the research problem is stated. The proposed control method is presented in Section 3. Other control methods are also adapted and applied to the macro-mini robotic scheme. They are presented in Section 4. A new prototype is introduced and the experiments are explained in Section 5. Experiments are carried on the new prototype and the performance of each of the control methods is compared. The advantages and drawbacks of each method are pointed out. Finally, some conclusions and future work are discussed in Section 6

Problem formulation

The macro-mini system consists of two manipulators connected in series. A large robot, referred to as the macro robot, is fixed to the base. Then, a small robot, referred to as the mini robot, is attached to the end-effector of the macro. The macro and the mini can have different numbers of degrees of freedom (DoF). However, the end-effector can perform $n$ -DoF tasks that combine the motion from the two robot components. If the mobility of each of the robots has a common intersection (e.g. each of the robots allow positioning dofs), then the macro-mini system has redundancy and is considered as a redundant manipulator. Without loss of generality, only the translational motion is considered in this work, in order to simplify the analysis. The kinematics of the macro-mini system is firstly analyzed. Then, considering several assumptions, a linear dynamic model of this combined system can be derived from the separate model of each robot component. Finally, the control problem is formulated.

Kinematic model of the macro-mini system

An arbitrary Cartesian work-frame ${\mathcal{R}}_w$ is defined and linked to the workspace of the macro-mini system so that all position vectors and trajectories can be expressed precisely and visually. The position vector of the macro’s end-effector is represented by ${\mathit{p}}_M\in {\mathbb{R}}^{3\times 1}$ , with respect to the work frame ${\mathcal{R}}_w$ . It is also assumed that the macro robot is a constant-orientation positioning robot, such as, for instance a gantry robot. In other words, the end-effector of the macro robot has a constant orientation, regarless of its position. A reference frame, namely ${\mathcal{R}}_m$ , is defined and attached to the base of the mini robot. ${\mathcal{R}}_m$ is arranged in a given orientation with respect to frame ${\mathcal{R}}_w$ described by a constant rotation matrix ${\mathit{Q}}_m\in {\mathbb{R}}^{3\times 3}$ . The position vector of the mini’s end-effector is defined as ${\mathit{p}}_m\in {\mathbb{R}}^{3\times 1}$ with respect to its reference frame ${\mathcal{R}}_m$ . This vector with respect to the work frame ${\mathcal{R}}_w$ , defined as ${\mathit{p}}_{\mathrm{Mm}}\in {\mathbb{R}}^{3\times 1}$ , is considered as the position vector of the macro-mini system. Controlling ${\mathit{p}}_{\mathrm{Mm}}$ is the main purpose of this work and it is determined in the following equation, constant offsets neglected.

$${\mathit{p}}_{\mathrm{Mm}}={\mathit{p}}_M+{\mathit{Q}}_m{\mathit{p}}_m.$$

(1)

Rewriting this equation in matrix form yields

$${\mathit{p}}_{\mathrm{Mm}}=\left[\begin{array}{cc}\mathit{I}\hfill & {\mathit{Q}}_m\hfill \end{array}\right]\left[\begin{array}{c}{\mathit{p}}_M\hfill \\ \multicolumn{1}{c}{{\mathit{p}}_m}\end{array}\right].$$

(2)

Taking the derivative of both sides with respect to time, the velocity equation is written as

$${\stackrel{·}{\mathit{p}}}_{\mathrm{Mm}}=\left[\begin{array}{cc}\mathit{I}\hfill & {\mathit{Q}}_m\hfill \end{array}\right]\left[\begin{array}{c}{\stackrel{·}{\mathit{p}}}_M\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{p}}}_m}\end{array}\right]$$

(3)

in which the Jacobian matrix of the macro-mini system is now defined as

$$\mathit{J}=\left[\begin{array}{cc}\mathit{I}\hfill & {\mathit{Q}}_m\hfill \end{array}\right].$$

(4)

Matrix $\mathit{J}$ represents the kinematics of the translation of the macro-mini system. It is constant and always full row rank and it depends only on the initial configuration. As a result, the combined system has no singularity.

Mathematical model and assumptions

First of all, the position of each robot is assumed to be managed by its own controller. More specifically, each robot is a standalone system including a trajectory tracking controller that functions independently. On each axis, the dynamics of each robot including the position controller are assumed to be expressed by a linear transfer function defined as

$$G\left(s\right)=\frac{P\left(s\right)}{P_r\left(s\right)}=\frac{K(b_0{s}^m+b_1{s}^{m-1}+...+1)}{a_0{s}^n+a_1{s}^{n-1}+...+1}(n>m)$$

(5)

in which K is a proportional gain, $b_0,b_1,...,a_0,a_1,...$ are constants. In other words, the reference position $p_r$ is now manipulated as the input and the real position of the robot $p$ is the output of this sub-system. In practice, $K$ should be approximately $1$ because the position controller of robots is designed so that the regulation error converges to zero. Also, the transfer function is often approximated as a first-order system or a second-order system.

In this article, ${\mathit{G}}_M\left(s\right),{\mathit{G}}_m\left(s\right)$ denote the multiple-input multiple-output (MIMO) transfer functions of the macro robot and the mini robot, respectively. Also, ${}^m{\mathit{G}}_M\left(s\right){,}^M{\mathit{G}}_m\left(s\right)\in {\mathbb{R}}^{3\times 3}$ represent the effect of the macro motion on the mini and vice versa. Therefore, the general model of a macro-mini system can be expressed as

$${\mathit{P}}_{\mathrm{Mm}}\left(s\right)=\mathit{J}\left[\begin{array}{cc}{\mathit{G}}_M\left(s\right)\hfill & {}^M{\mathit{G}}_m\left(s\right)\hfill \\ \multicolumn{1}{c}{{}^m{\mathit{G}}_M\left(s\right)}& {\mathit{G}}_m\left(s\right)\hfill \end{array}\right]\left[\begin{array}{c}{\mathit{P}}_{\mathrm{rM}}\left(s\right)\hfill \\ \multicolumn{1}{c}{{\mathit{P}}_{\mathrm{rm}}\left(s\right)}\end{array}\right].$$

(6)

However, the mechanical design of the robots can produce independent motion on each axis and the inner control loops are assumed to function precisely. Then, the motion of one axis is not affected by that of the others. Thus, ${\mathit{G}}_M\left(s\right),{\mathit{G}}_m\left(s\right)$ are diagonal matrices. The motion of the macro does not significantly affect the dynamics of the mini because the macro can produce only limited accelerations and the mini position controller is very stiff. As a result, one has ${}^m{\mathit{G}}_M\left(s\right)=0_{3\times 3}$ . On the other hand, the relative ratio between the moving mass of the macro and the moving mass of the mini is high. It is possible that gear boxes are used and that high friction can be seen on all axes of the macro. Therefore, the mini’s inertia force is not high enough to have any effect on the macro outputs. This fact leads to ${}^M{\mathit{G}}_m\left(s\right)=0_{3\times 3}$ . Even so, the total mass of the mini is mounted on the end-effector of the macro and ${\mathit{G}}_M\left(s\right)$ is not the same as the macro working alone and this is taken into account during the identification process. Eventually, a simplified model of the macro-mini is written as

$${\mathit{P}}_{\mathrm{Mm}}\left(s\right)=\mathit{J}\left[\begin{array}{cc}{\mathit{G}}_M\left(s\right)\hfill & 0_{3\times 3}\hfill \\ \multicolumn{1}{c}{0_{3\times 3}}& {\mathit{G}}_m\left(s\right)\hfill \end{array}\right]\left[\begin{array}{c}{\mathit{P}}_{\mathrm{rM}}\left(s\right)\hfill \\ \multicolumn{1}{c}{{\mathit{P}}_{\mathrm{rm}}\left(s\right)}\end{array}\right]$$

(7)

in which ${\mathit{G}}_M\left(s\right),{\mathit{G}}_m\left(s\right)$ can be divided into single-input single-output (SISO) transfer functions: $G_M^x\left(s\right),G_M^y\left(s\right),G_M^z\left(s\right)$ and $G_m^x\left(s\right),G_m^y\left(s\right),G_m^z\left(s\right)$ . Each of them is then identified experimentally. In the rest of this article, the macro-mini model is described by equation (7) and laid out in Figure 1.

Figure 1.

Diagram of the simplified macro-mini model.

Hence, a simplified mathematical model of the macro-mini system can be derived from the separate model of each robot, which can be easier to determine or simpler to identify. This study applies the cascade model where the proposed control method works at the task-level, on top of the position controller of the macro and the mini. Mid-ranging control and model predictive control are also adapted to apply to this control scheme.

Proposed control method: Combination of PI compensation and redundancy resolution (PI+RR)

In general, the relationship between the velocity in the joint coordinates and that in the task coordinates is expressed as

$$\stackrel{·}{\mathit{x}}=\mathit{J}\stackrel{·}{\mathit{q}}$$

(8)

in which $\mathit{J}$ is the Jacobian matrix of the manipulator. In this case, $\mathit{J}$ is the Jacobian matrix of the macro-mini defined in section 2.1, $\mathit{x}={\mathit{p}}_{\mathrm{Mm}}$ and $\mathit{q}=[{\mathit{p}}_M^T,{\mathit{p}}_m^T{]}^T$ . As mentioned above, the macro-mini system is considered as a redundant manipulator. Considering equation (8) in a short time period, the resolved motion rate problem ²³ can be applied to compute a small variation $\delta \mathit{q}$ from a given $\delta \mathit{x}$ and the current joint coordinates $\mathit{q}$ . The general form is given as follows

$$\delta \mathit{q}={\mathit{J}}^{†}\left(\mathit{q}\right)\delta \mathit{x}+[\mathit{I}-{\mathit{J}}^{†}\left(\mathit{q}\right)\mathit{J}\left(\mathit{q}\right)]\mathit{z}$$

(9)

in which ${\mathit{J}}^{†}$ is the generalized inverse of $\mathit{J}\left(\mathit{q}\right)$ , $\mathbf{z}$ is an arbitrary vector and $\mathit{I}$ stands for the indentity matrix. Differently from the inverse kinematics, which are nonlinear and not a bi-directional mapping between two spaces, the resolved motion rate gives a unique solution at each iteration. Typically, Jacobian matrix $\mathit{J}$ is a constant matrix because the configuration of the macro-mini has been fixed. Also, singularities can be completely avoided in this system. Thence, the generalized inverse ${\mathit{J}}^{†}$ and vector $\mathbf{z}$ need to be determined to suitably distribute the displacement between the two robots while taking into account their proper dynamics and physical limits. These problems are addressed in the following sections.

Weighted generalized inverse and the dynamical consistency

As mentioned above, the dynamics of the macro and the mini robots are drastically different. In order to take this factor into account while determining the generalized inverse ${\mathbf{J}}^{†}$ , the following optimization problem is considered.

$$\begin{array}{c}\min \mathit{K}=\frac{1}{2}\delta {\mathit{q}}^T\mathit{W}\delta \mathit{q}\hfill \\ \mathrm{subject}\mathrm{to}\delta \mathit{x}=\mathit{J}\delta \mathit{q}\hfill \end{array}$$

(10)

where $\mathit{K}$ represents the total kinetic energy of the system provided that a proper adjustable weighting matrix $\mathit{W}$ is selected. This problem consists in finding $\mathit{q}$ , which satisfies the given mapping, so that it minimizes the total kinetic energy of the system. The general solution to this problem based on the weighted generalized inverse is given as

$$\delta \mathit{q}={\mathit{J}}_w^{†}\delta \mathit{x}+[\mathit{I}-{\mathit{J}}_w^{†}\left(\mathit{q}\right)\mathit{J}\left(\mathit{q}\right)]\mathit{z}$$

(11)

where

$${\mathit{J}}_w^{†}={\mathit{W}}^{-1}{\mathit{J}}^T{\left[\mathit{J}{\mathit{W}}^{-1}{\mathit{J}}^T\right]}^{-1}.$$

(12)

Weighting matrix $\mathit{W}$ reflects the dynamical differences of the two robot components in the equation. The result can be interpreted as the division of the desired input into the desired displacement for the macro and the mini. However, the dynamics of linear systems, which include the transient response and the steady-state response, cannot be generally described by a constant. In this study, the transient response is focused on and the cut-off frequency is selected. On the one hand, this frequency relatively describes the speed of response. ²⁴ On the other hand, the steady-state error is supposed to be approximately zero and will be improved later. This parameter can also be determined experimentally based on the frequency response of any linear system. $\mathit{W}$ is normalized as

$$\mathit{W}=\left[\begin{array}{cc}\left[\begin{array}{ccc}\frac{{\omega }_{\mathrm{cm}}^x}{{\omega }_{\mathrm{cM}}^x}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{{\omega }_{\mathrm{cm}}^y}{{\omega }_{\mathrm{cM}}^y}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \frac{{\omega }_{\mathrm{cm}}^z}{{\omega }_{\mathrm{cM}}^z}\hfill \end{array}\right]\hfill & 0_{3\times 3}\hfill \\ \multicolumn{1}{c}{0_{3\times 3}}& {\mathit{I}}_{3\times 3}\hfill \end{array}\right]$$

(13)

where ${\omega }_{\mathrm{cm}}^x,...,{\omega }_{\mathrm{cM}}^x,...$ are cut-off frequencies of the mini and the macro on the corresponding axis $x,y$ and $z$ .

In Ref., ²⁵ the authors showed that the generalized inverse matrix that is consistent with the dynamics of the robot for non-redundant and redundant manipulators is

$${\mathit{J}}^{*}={\mathit{M}}^{-1}{\mathit{J}}^T{\left[\mathit{J}{\mathit{M}}^{-1}{\mathit{J}}^T\right]}^{-1}.$$

(14)

in which $\mathit{M}$ is the inertia matrix of the manipulator. It can be seen from equations (12) and (14) that the weighting matrix $\mathit{W}$ plays the role of the simplified inertia matrix of the macro-mini system. Further investigation can be pursued to describe more closely the dynamics of this system.

Penalty function and physical limits

Beside the differences in the dynamics, the physical limits of the macro and the mini are also very different. The mini has a very small workspace compared to that of the macro. As a result, the proposed controller needs to slow down the mini whenever it approaches to these ends. A penalty function $\mathcal{L}$ is specifically defined based on workspace constraints. This function, $\mathcal{L}$ , has the following characteristics

• (i) $\mathcal{L}$ is a scalar function.

• (ii) $\mathcal{L}$ is strictly convex.

$$\begin{array}{c}\forall {\mathit{p}}_1,{\mathit{p}}_2\in \mathbb{W}:\mathcal{L}(ϵ{\mathit{p}}_1+(1-ϵ){\mathit{p}}_2)<ϵ\mathcal{L}\left({\mathit{p}}_1\right)\hfill \\ +(1-ϵ)\mathcal{L}\left({\mathit{p}}_2\right)(\forall ϵ\in \mathbb{R}|ϵ<1).\hfill \end{array}$$

• (iii) The minimum of $\mathcal{L}$ is located at the center of the workspace.

$$\begin{array}{c}{\mathit{p}}_O=\frac{{\mathit{p}}_{\min }+{\mathit{p}}_{\max }}{2}={\arg }_{\forall \mathit{p}\in \mathbb{W}}\mathcal{L}\left(\mathit{p}\right).\hfill \end{array}$$

The problem of handling the physical limits turns into obtaining the nullspace solution, which is reflected on vector $\mathit{z}$ , in order to minimize this function. There is no unique choice of function that satisfies the properties listed above in the workspace $\mathbb{W}$ of the macro-mini system. Then, vector $\mathit{z}$ is chosen so that it points in the inverse direction of the gradient vector, namely

$$\mathit{z}=-\mathit{L}(\nabla \mathcal{L})$$

(15)

where a constant diagonal matrix $\mathit{L}$ is added to adapt the behavior of the controller. In practice, the choice of this function is rather critical because the constraint handling depends very much on this choice. From simulations and experiments, the magnitude of vector $\mathit{z}$ should be approximately zero when the robot stays near the origin so that the nullspace solution does not adversely affect the generalized inverse solution. Then, the magnitude of vector $z$ increases whenever the robots approach their limits on any axis from a safe distance. The nullspace solution ensures that the two robots respect workspace limits as long as the trajectory is chosen reasonably.

Implementation

In order to implement equation (11), it is rewritten in a discrete form using the backward difference approximation,

$$\begin{array}{c}\delta \mathit{q}(k+1)=\frac{{\mathit{q}}_r(k+1)-\mathit{q}\left(k\right)}{T_s}\hfill \\ \delta \mathit{x}(k+1)=\frac{{\mathit{x}}_r(k+1)-\mathit{x}\left(k\right)}{T_s}\hfill \end{array}$$

An iterative procedure can be obtained for this system as

$${\mathit{q}}_r(k+1)=\mathit{q}\left(k\right)+{\mathit{J}}_w^{†}\delta \mathit{x}\left(k\right)+T_s(\mathit{I}-{\mathit{J}}_w^{†}\mathit{J})\mathit{z}\left(k\right)$$

(16)

in which the feedback positions $\mathit{q}\left(k\right),\mathit{x}\left(k\right)$ are applied in this approximation instead of the previous reference signals ${\mathit{q}}_r\left(k\right),{\mathit{x}}_r\left(k\right)$ in order to avoid the accumulated position error.

Lastly, a PI compensation can be added and acts as a low-pass filter to keep the input signal from being too aggressive. The output signal from this controller is fed into the redundancy resolution. Moreover, the PI component also eliminates the steady-state error, which happens when the dynamics of the system are not modeled perfectly. The schematic of the proposed control method is finally illustrated in Figure 2.

Figure 2.

Diagram of the proposed control method (PI + RR).

Adaptation of other control methods to multiple-DoF macro-mini systems

Other control methods can be adapted into the cascade layout in this work. Therefrom, the performance of the proposed control can be compared. In this work, mid-ranging control and model predictive control are selected as respectively presented in the following sections.

Mid-ranging control

Valve position controller (VPC) was investigated in Ref. ²⁶ in order to regulate Multiple-Input Single-Output (MISO) systems using SISO control methods. Mid-ranging control is a modified version of VPC and it is analyzed in more detail in Ref. ²⁷ A simulation analysis of this control method for a macro-mini manipulator was studied in Ref. ²⁸ It was also adapted to a 1-DoF macro/micro actuation in Ref. ¹⁸ In this work, this method is generalized for multiple redundant DoF systems and the new scheme is shown in Figure 3.

Figure 3.

Diagram of mid-ranging controller.

Similarly to the conventional scheme, ${\mathit{G}}_{c1}$ is composed of PI controllers and ${\mathit{G}}_{c2}$ of integral controllers. In a modified version introduced in Ref., ²⁶ ${\mathit{G}}_{c2}$ can also be composed of PI controllers. PI and I controllers have similar responses in a low frequency range, while their behavior is very different at high frequencies. A PI controller allows the transmission of some high-frequency signals thanks to the P-parameter, while I-controller filters out all these signals. A proper choice has to rely on a specific control purpose. The macro here cannot deal with any high frequency signals and therefore an I-controller is chosen. Adjusting the integral gain causes a change in the cut-off frequency value.

$$G_{c1}^k=K_{p1}+\frac{K_{i1}}{s},G_{c2}^k=\frac{K_{i2}}{s}(k=x,y,z).$$

(17)

After doing this modification, the macro-mini system is very much alike a combination of independent SISO systems, for which the plant is represented by ${\mathit{G}}_{\mathrm{IMm}}\left(s\right)$ as follows (instead of equation (7))

$${\mathit{G}}_{\mathrm{IMm}}\left(s\right)=\mathit{J}\left[\begin{array}{cc}{\mathit{G}}_{c2}{\mathit{G}}_M\left(s\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{G}}_m\left(s\right)\hfill \end{array}\right]{\mathit{J}}^T.$$

(18)

A classical problem is established on each axis where the parameters of $G_{c1}^x,G_{c1}^y$ or $G_{c1}^z$ can be computed by approximation as in Ref. ²⁶ or algebraically for linear systems as in Ref. ²⁹

Model predictive control

From the transfer function in equation (7), a discrete state space model can be derived at a particular sampling time $T_s$ , which can be written as

$$\{\begin{array}{c}\mathit{x}\left(k\right)=\mathit{Ax}(k-1)+\mathit{Bu}\left(k\right)\hfill \\ \multicolumn{1}{c}{\mathit{y}\left(k\right)=\mathit{Cx}\left(k\right)}\end{array}$$

(19)

in which $\mathit{x}\left(k\right)=[{\mathit{p}}_M^T\left(k\right),{\mathit{p}}_m^T\left(k\right){]}^T,\mathit{u}\left(k\right)=[{\mathit{u}}_M^T,{\mathit{u}}_m^T{]}^T=[{\mathit{p}}_{\mathrm{rM}}^T,{\mathit{p}}_{\mathrm{rm}}^T{]}^T$ and $y\left(\mathit{k}\right)={\mathit{p}}_{\mathrm{Mm}}$ . Since the positions of the mini and the macro are all available and the model of this system is linear, full state feedback would probably be a promising method. Based on some optimization, this linear control method can rationally govern each subsystem to satisfy their dynamic characteristics. However, it turns out that this method cannot handle constraints, especially inequality, hard constraints as can be seen in this case. The physical limits of the macro and the mini robots were considered quite simply in the previous method by a penalty function. Besides, both previous methods cannot take into account some prior information about the reference path which is often available and predefined. The model-based control method, Model Predictive Control (MPC), is well fit to improve these weaknesses. The fundamental mathematics of this method were developed in many works, for example Ref. ³⁰ In this section, the generalized predictive control (GPC) using a state-space model is applied and control parameters are adapted for the current system as examined below.

The limited workspace of the two robots is represented by inequality constraints. Assuming a symmetrical workspace, one can write

$$\begin{array}{c}\left|{\mathit{u}}_M\left(k\right)\right|\le {\mathit{p}}_{\mathrm{Mmax}}\hfill \\ \multicolumn{1}{c}{\left|{\mathit{u}}_m\left(k\right)\right|\le {\mathit{p}}_{\mathrm{mmax}}.}\end{array}$$

(20)

Along with these restrictions, the macro and the mini are also limited by maximum velocities, which can be written as

$$\begin{array}{c}\left|\delta {\mathit{u}}_M(k+1)\right|=|{\mathit{p}}_{\mathrm{rM}}(k+1)-{\mathit{p}}_{\mathrm{rM}}\left(k\right)|\le T_s{\mathit{v}}_{\mathrm{Mmax}}\hfill \\ \left|\delta {\mathit{u}}_m(k+1)\right|=|{\mathit{p}}_{\mathrm{rm}}(k+1)-{\mathit{p}}_{\mathrm{rm}}\left(k\right)|\le T_s{\mathit{v}}_{\mathrm{mmax}}.\hfill \end{array}$$

(21)

Based on equation (19), at a specific time step $k$ , if control actions $\mathit{u}(k+i)$ are applied, future states $\mathit{x}(k+i|k)$ and future outputs $\mathit{y}(k+i|k)$ of the system can be predicted and denoted by

$$\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}\mathit{X}\left(k\right)=[\mathit{x}(k+1|k)\mathit{x}(k+2|k)...\mathit{x}(k+N_p|k)]\hfill \\ \multicolumn{1}{c}{\mathit{Y}\left(k\right)=[\mathit{y}(k+1|k)\mathit{y}(k+2|k)...\mathit{y}(k+N_p|k)]}\end{array}}\end{array}$$

where $N_p$ is called the prediction horizon and can be varied. In order to follow the trajectory, the control signal $\mathit{u}\left(k\right)$ is manipulated at each step by a change $\delta \mathit{u}(k+1)=\mathit{u}(k+1)-\mathit{u}\left(k\right)$ . The control strategy calculated at the moment $k$ is noted by a series of these changes.

$$\Delta \mathit{U}\left(k\right)=[\delta \mathit{u}(k+1)\delta \mathit{u}(k+2)...\delta \mathit{u}(k+N_c)]$$

where $N_c\le N_p$ is called the control horizon and can also be varied. In case $N_c<N_p$ , input change of the remaining elements from $N_c$ to $N_p$ is equal to 0, that is, $\mathit{u}(k+i)=\mathit{u}(k+i-1)(N_c<i\le N_p)$ .

The MPC determines a unique optimal control sequence $\delta {\mathit{u}}^{*}\left(k\right)$ by minimizing a cost function $\mathcal{V}$ while taking into account all the constraints defined in equations (20) and (21). The cost function is determined based on control objectives considering the relevant inputs and outputs. For example, in this case

$$\mathcal{V}=\sum _{j=x,y,z}\left\{\sum _{i=1}^{i=N_p}{\left[p_{rMm}^j\left(k+i|k\right)-p_{Mm}^j\left(k+i|k\right)\right]}^2+\sum _{i=1}^{i=N_c}{\lambda }_{Mi}^j{\left[\delta u_M^j\left(k+i\right)\right]}^2+\sum _{i=1}^{i=N_c}{\lambda }_{mi}^j{\left[\delta u_m^j\left(k+i\right)\right]}^2+\sum _{i=1}^{i=N_c}{\varphi }_{mi}^j{\left[u_m^j\left(k+i\right)\right]}^2\right\}$$

(22)

in which ${\lambda }_{\mathrm{Mi}}^x,{\lambda }_{\mathrm{mi}}^x$ are normalized and tunable weight constants corresponding to the macro input and the mini input on the $x$ -axis at step $i$ . They indicate the relative importance of input changes compared to the main control purpose: following the trajectory. Similarly, ${\varphi }_i$ are other normalized weight constants which also denote the relative importance of keeping the mini input values near zero. In this case, mid-ranging the macro inputs is not a priority. ${\lambda }_i,{\varphi }_i$ can be chosen differently for each future control action. In general, there is always a compromise between the control accuracy and other requirements such as respecting speed limits, mid-ranging the mini position, etc. Solving similar optimal problems has been studied in many researches, ³⁰ as an example.

Generally speaking, MPC takes into consideration all factors that affect the performance. In the meantime, the need to adjust several parameters simultaneously in order to achieve this control objective is a challenge. As also mentioned in Ref., ³⁰ tuning these parameters still requires trial-and-error with some acquired experience. There is no systematic or analytical method for this tuning process. However, the output changes can be partially predicted. In this case, ${\varphi }_i$ can be increased by the end and should be very small compared to the other parameters because keeping the mini near the origin is an auxiliary objective. Figure 4 shows an outline of this control method.

Figure 4.

Diagram of MPC control.

Experimental results

Different experiments are carried out in this section on a new design of macro-mini robot, in which a 3-DoF gantry robot is coupled to a kinematically redundant (6+3)-DoF parallel robot proposed in Ref. ³¹ The schematic of the new robotic system is illustrated in Figure 5(a).

Figure 5.

Experimental setup: (a) schematic of the macro-mini system, (b) hardware block diagram, (c) macro-mini system, and (d) the mini robot.

On the macro side, the actuator on each axis is a Parker brushless servo motor MPJ1153C with a gearbox 7:1 in the $x,y$ direction, 20:1 in the $z$ direction. The motors are driven in the torque mode. The position of each joint is measured using an incremental encoder of 8000 pulses/turn. On the other hand, the nine mini actuators are Maxon motor EC 90 flat, brushless, 260 W without any gearbox. This direct-drive architecture enables the robot to be backdrivable and reduces nonlinear effects such as friction and backlash. In the meantime, each active joint includes an incremental encoder with 16,384 counts per turn in quadrature mode. All the motors are driven by Digital Servo Drive from Elmo in the Torque Control Mode. The controllers are implemented in the Simulink Real-Time environment, which is operated on a Target PC (Intel Core-i5, 4 cores 3.1 GHz, 8 GB RAM). The sampling time is set at $T_s=1$ (ms) thanks to the high-speed communication protocol, EtherCAT. It ensures a stable connection between the drivers and the PC. A hardware block diagram is shown in Figure 5(b).

The mini robot consists of a moving platform and three fully actuated legs. This architecture supports 3 DoFs in translation, 3 DoFs in rotation and 3 other redundant DoFs. Regarding the number of redundant DoFs with the macro robot, only three translational DoFs are considered for the redundant control of the macro-mini system. The other DoFs are unique to the mini and used as an extension. The total mass and the moving mass of this robot are respectively $16\left(\mathrm{kg}\right),11.2\left(\mathrm{kg}\right)$ , but most of the moving components are placed near the base. The motion controller of this mini robot was developed and presented in Ref. ³² As experimented in this study, this robot has the capability to produce accelerations of up to 10 $m/s^2$ , which is very high compared to the gantry. On the other hand, its motion is narrowed in a range of $0.25m\times 0.25m\times 0.2m$ in the directions of $x,y,z$ which is obviously very small compared to the workspace of the gantry. Hence, the two robots correspond perfectly to the concept of the macro-mini system discussed in the previous section. In the meantime, the macro robot and its position controller are also described in more detail in Appendix A. The redundancy of this system is demonstrated in the following video (https://youtu.be/f-9\_\_ooY86U).

The linear model of each sub-system, namely the macro robot and the mini robot, is identified and presented in Appendix B. Considering the combination of the two robots in $x,y$ directions, the weighting matrix $\mathit{W}$ is defined as follows

$$\mathit{W}=\left[\begin{array}{cc}5{\mathbf{I}}_{2\times 2}\hfill & 0_{2\times 2}\hfill \\ \multicolumn{1}{c}{0_{2\times 2}}& {\mathit{I}}_{2\times 2}\hfill \end{array}\right]$$

(23)

In most cases, the workspaces of the macro and the mini are symmetrical, that is, ${\mathit{p}}_{\max }=-{\mathit{p}}_{\min }$ , and this condition also applies in our work. Therefore, the penalty function $\mathcal{L}$ and vector $\mathbf{z}$ are specified in this work as

$$\mathcal{L}=\sum _{i=x,y}[\frac{1}{({p_{\mathrm{Mmax}}^i}^2-{p_M^i}^2)}+\frac{1}{({p_{\mathrm{mmax}}^i}^2-{p_m^i}^2)}]$$

(24)

$$\mathit{z}=-\mathit{L}{\left[\begin{array}{cccc}\frac{\partial \mathcal{L}}{\partial p_M^x}\hfill & \frac{\partial \mathcal{L}}{\partial p_M^y}\hfill & \frac{\partial \mathcal{L}}{\partial p_m^x}\hfill & \frac{\partial \mathcal{L}}{\partial p_m^y}\hfill \end{array}\right]}^T$$

(25)

The value of the penalty function and the negative gradient vector in the one dimensional case are given below and illustrated in Figure 6.

$$\mathcal{L}=\frac{1}{({p_{\mathrm{Mmax}}^x}^2-{p_M^x}^2)}+\frac{1}{({p_{\mathrm{mmax}}^x}^2-{p_m^x}^2)}$$

(26)

$$-\nabla \mathcal{L}=-{\left[\begin{array}{cc}\frac{2p_M^x}{{({p_{\mathrm{Mmax}}^x}^2-{p_M^x}^2)}^2}\hfill & \frac{2p_m^x}{{({p_{\mathrm{mmax}}^x}^2-{p_m^x}^2)}^2}\hfill \end{array}\right]}^T$$

(27)

Figure 6.

An illustration of the penalty function on the $x$ -axis: (a) penalty function value and (b) gradient vector field.

The setup for the experiment in this work follows the preferred arrangement in which ${\mathcal{R}}_m$ and ${\mathcal{R}}_w$ are aligned. More specifically, ${\mathit{Q}}_m$ is equal to the identity matrix ${\mathit{I}}_{3\times 3}$ . An overview of the experimental setup is shown in Figure 5(c) and (d). Control parameters are given in Table 1.

Table 1.

Control parameters in the experiments.

Mid-ranging	$K_{p1}=8,K_{i1}=3,K_{i2}=4$
RR + PI	$\begin{array}{c}{K}_p=8,K_i=3,L=12\hfill \\ p_{\mathrm{Mmax}}\hfill & =1100\left(\mathrm{mm}\right)\hfill \\ p_{\mathrm{mmax}}\hfill & =220\left(\mathrm{mm}\right)\hfill \end{array}$
MPC	$\begin{array}{c}\left|p_M\right|\le 1000(+100)\left(\mathrm{mm}\right)\hfill \\ \left|p_m\right|\le 200(+20)\left(\mathrm{mm}\right)\hfill \\ \left|v_M\right|\le 300(+10)(\mathrm{mm}/s)\hfill \\ \left|v_m\right|\le 1000(+100)(\mathrm{mm}/s)\hfill \\ N_p=N_u=10\hfill \\ {\lambda }_{\mathrm{Mi}}=0.35\hfill \\ {\lambda }_{\mathrm{mi}}=0.025\hfill \\ {\varphi }_{\mathrm{mi}}=0.05\hfill \end{array}$

In the first set of experiments, the robot performs the same trajectory along the $x$ -axis applying each of the control methods in order to demonstrate and to evaluate the proposed approach more clearly. Afterward, a demo that combines multiple axes is investigated to show an application of the new macro-mini system. The control performance is examined by considering the mean absolute error (MAE), the maximum error (MaxE) and the percent overshoot ( $P.O.$ , similarly to the a performance index of linear systems where $p_{\mathrm{ss}}$ is the final steady state value). These indices are written as

$$\begin{array}{l}MAE=\frac{\sum ^{\left|e\left(k\right)\right|}}{N},MaxE=max[e\left(k\right)],\\ P.O.=\frac{max\left(p\right)-p_{ss}}{p_{ss}}\times 100\%.\end{array}$$

Simulation and experiment on the controller behavior in the frequency domain

The first trajectory is the sum of two sine waves with different frequencies given as

$$s\left(t\right)=A_1\sin {\omega }_{1}t+A_2\sin {\omega }_{2}t$$

(28)

in which $A_1=50\left(\mathrm{mm}\right),A_2=500\left(\mathrm{mm}\right)$ are the magnitudes of sine waves, ${\omega }_1=8(\mathrm{rad}/s),{\omega }_2=0.5(\mathrm{rad}/s)$ are their frequencies. As a result, this trajectory can not only verify the control performance of each control method but also show more clearly their behavior in the frequency domain, which is important as well. Each control method is applied. Then, the positions of the macro, of the mini and of the combined macro-mini are logged and analyzed.

Firstly, the tracking performance of the macro-mini system using the three control methods is experimented and the results are shown in Figure 7. The tracking error of each approach is examined and compared as summarized in Table 2 regarding MAE and MaxE. The proposed method PI + RR demonstrates a performance similar to that of the mid-ranging control in this comparison. Also, MPC shows improvements on both indices compared to the others.

Figure 7.

The macro-mini position response in three experiments with the control methods.

Table 2.

Comparison of MAE and MaxE of the three control methods in the experiment with the combined sinewaves.

Control method	MAE (mm)	MaxE (mm)
Mid-ranging	4.5654	17.3891
PI + RR	4.5882	15.6590
MPC	3.5816	12.4975

Furthermore, the same desired signal is used in simulations. The linear model of equation (7) and each of the three control methods are respectively applied. The motion of each robot component, namely the macro and the mini, during the trajectory from the simulations is compared to that from the experiments as can be seen in Figure 8 for mid-ranging control, in Figure 9 for PI + RR and in Figure 10 for MPC. It can be noted that the simulations and the experiments return similar results. On one hand, the results validate the linear model in Appendix B. On the other hand, the three control methods in the task space demonstrate consistent responses in the frequency domain, in which the macro is responsible for low frequency motion and the mini handles high frequency motion. The same behavior is achieved in different ways, either with a simple linear controller or a complex optimal controller. These results show that dividing tasks based on frequency response is an appropriate strategy for the macro-mini system. The control methods are then tested on a the second trajectory for their different behavior in handling constraints. Finally, since the results of the simulations and experiments are very similar, the parameters can be tuned based on simulation, even though this may not lead to the optimal set of parameters in practice.

Figure 8.

Macro position and Mini position in a simulation and an experiment with Mid-Ranging Control.

Figure 9.

Macro position and Mini position in a simulation and an experiment with PI + RR.

Figure 10.

Macro position and Mini position in a simulation and an experiment with MPC.

Experiments on constraint handling and mid-ranging of the mini

The second trajectory consists of two point-to-point motions. The robot is commanded to go from $x=0$ to $x=500\left(\mathrm{mm}\right)$ , the center of the workspace, and to $x=1000\left(\mathrm{mm}\right)$ , the limit of the macro. Both distances are certainly out of the mini’s range. They are simply named P2P-500 and P2P-1000, respectively. The trajectories are programed to follow the fifth-order polynomial function described by

$$s\left(\tau \right)=6{\tau }^5-15{\tau }^4+10{\tau }^3$$

(29)

in which $\tau =\frac{t}{T}$ , where $t$ is thetime and $T$ is the period of the trajectory. The desired position can be tracked closely using either one of the three methods as illustrated in Figure 11 for the P2P-500 trajectory. Similar responses are obtained for the P2P-1000 trajectory in Figure 12. However, considering the tracking error, their performance is compared again in Tables 3 and 4 based on MAE, MaxE and $P.O.$ . As opposed to the previous experiment, the proposed method PI + RR shows a better result on all three indices compared to mid-ranging control. It significantly reduces the overshoot caused by the integral term of mid-ranging control, $G_{c2}$ , as discussed before. Once again, MPC demonstrates much better performance, which notably reduces the tracking error and the overshoot.

Figure 11.

The macro-mini position response in an experiment point-to-point 500 mm.

Figure 12.

The macro-mini position response in an experiment point-to-point 1000 mm.

Table 3.

Comparison of MAE, MaxE and P.O. of the three control methods in the experiment point-to-point 500 mm.

Control method	MAE (mm)	MaxE (mm)	P.O. (%)
Mid-ranging	5.6856	11.0869	2.22
PI + RR	4.2781	9.7193	0.84
MPC	1.3102	4.0579	0.81

Table 4.

Comparison of MAE, MaxE and P.O. of the three control methods in the experiment point-to-point 1000 mm.

Control method	MAE (mm)	MaxE (mm)	P.O.(%)
Mid-ranging	5.4505	14.0606	1.07
PI + RR	5.1117	9.5805	0.65
MPC	1.0064	2.7441	0.17

The motion of the macro and the mini obtained with the three different controllers is recorded and shown in Figures 13 and 14. In this case, the macro and the mini demonstrate different reactions when the limits are approached. In the first point-to-point experiment, the three control methods show a similar behavior. In fact, the three methods can relatively mid-range the mini. For PI + RR, the results depend on the definition of the penalty function, while for MPC, the results depend on how the mid-range condition is compared to the other control objectives. More interestingly, the differences in handling limits are shown in the second point-to-point experiment, where the macro limit is also reached. Mid-ranging control certainly cannot take any constraint into consideration. PI + RR tries to balance the limits of the macro and the mini so that no limit is reached. Therefore, the mini does not return to its origin. Lastly, MPC performs normally as long as no constraint is violated.

Figure 13.

Point to Point experiment from $x=0$ to $x=500$ mm.

Figure 14.

Point to Point experiment from $x=0$ to $x=1000$ mm.

Demonstration of the multiple-DoF combination

Control performance has been analyzed in the previous section. One of the control methods, namely MPC, is selected and the implementation of the proposed method is extended to multiple dimensions (position in $x$ and $y$ in this case) in a specific demo. Also, additional DoFs (rotational and redundant) of the mini are included. The extension allows to execute more complex tasks in which translation, rotation and grasping need to be coordinated fluently together, which is the motivation for implementing macro-mini systems. Because the motion along the $z$ -axis needed for the pick-and-place is limited to a short range that the mini can handle, translational motion in the $x$ and $y$ directions in the demo are accomplished by the combined system, while motion in the $z$ direction, rotational motion and grasping are done by the mini. More details on the motion control and grasping force control of the mini robot can be found in these references:.^32,33 Nevertheless, the same procedure can be applied to combine the motion on z-axis of the two robots.

More detail can be seen from the recorded video that accompanies this article (https://youtu.be/pZClOVh\_lPo). Wooden blocks are picked and placed so that they finally form the word “ROBOT.” This task includes nearly all possible motions: short-range translation, long-range translation, rotation about the $z$ -axis and opening/closing the gripper using the redundant DoFs. Due to uncertainties, some trial and error iterations were required to precisely adjust the trajectory so that it matches the real kinematics of the robot.

Discussion

The mid-ranging control is quite straightforward to apply. On the positive side, the bandwidth of the combined system is clearly extended as PI parameters are increased. Also, when PI parameters are high enough, control error is reduced and becomes much better compared to that of either the macro or the mini. However, the closed-loop system is controlled based on feedback signals, which are the current position of the macro and the mini. Thus, measurement noise and external disturbances are reflected on the control signal or even amplified. This factor causes vibrations and limits the value of the proportional gain Kp. Even though mid-ranging control is relatively simple and yet effective in many cases, it should not be recommended in some specific situations where tight constraints must be satisfied. As long as the mathematical model is precise, MPC is a powerful tool that gives the flexibility to improve control performance and handle constraints. Nevertheless, many parameters need to be adjusted, which may be a challenge. It also requires extensive computing capability as the number of DoF is increased.

PI + RR is proposed as a compromise in this context. The method mainly consists of matrix multiplications. Therefore, it is simple to implement and it does not require much computing power compared to MPC. Constraints, like physical limits, can be basically handled, while no prior information of the desired trajectory is required. Adversely, it can make the system become nonlinear near the physical limits and the nonlinearity depends much on the choice of the penalty function. Thence, each method presented above has its own advantages and drawbacks.

Conclusion

In this article, a new architecture of macro-mini system is introduced. After that, a mathematical model of the system is formalized. Some analysis is verified in practice in order to simplify this model. A new method combining PI and the redundancy resolution is proposed. Mid-ranging control and MPC are adapted and generalized for multi-DoF redundant macro-mini systems. The control architecture proposed in this work enables to simply generate trajectories in the task space, which is clearer and more visual. In addition to the theoretical analysis, each control method can take advantage of available MATLAB toolboxes in examining and tuning parameters. Then, simulations and experiments have been carried out. Firstly, the identified model and the task-space approach were validated. Secondly, the performance of the three control methods can be compared. In the frequency domain, the three controllers yield similar behaviors, in which the macro handles low frequencies of the desired input and the mini is responsible for the high frequencies. The experimental results also demonstrate the better ability of the proposed method to reduce the control error and the overshoot compared to the mid-ranging control but not as significantly as MPC. Additionally, the workspace limits can be simply included based on the definition of a penalty function, while this type of constraints cannot be considered in the mid-ranging controller. On the other hand, the proposed method requires much less computing power than MPC and does not need prior information, for example $(N_p-1)$ future steps of the reference signal, like MPC. Therefore, it is more applicable in many situations and can be extended for robotic systems with a large number of DoFs.

In practice, the proposed approaches are relatively sensitive to measurement noise, which affects more seriously mid-ranging control and PI + RR. A Kalman filter was applied to reduce this effect. Additional sensors can be used in the future so that the effect of noise can be weakened in the feedback signals. Besides, the combined system is assumed to be decoupled. This hypothesis is to simplify the dynamic model at low frequencies but it is not completely true. Small coupling still exists and can be observed in the identification process. The mini position is not absolutely zero at high frequencies in the green rectangle. Furthermore, the nonlinearity of the proposed method PI + RR, as well as of the mid-ranging control and MPC, near the limits will also need further examinations in more detail in future works. The weighting matrix is constant in this work but it can be variable and frequency-dependent to better represent the dynamical relationship between the macro and the mini.

Finally, a further investigation into the coupling terms ${}^m{\mathit{G}}_M\left(s\right)$ and ${}^M{\mathit{G}}_m\left(s\right)$ can lead to higher precision control methods. The uncertainty of identified models can be analyzed and it helps to generally examine the stability and robustness of each control method, as well as to enhance their performance. The nonlinearity near physical limits needs to be analyzed in more detail to avoid undesirable effects. Mobile robots on which robotic arms are mounted can be considered as a macro-mini system and the task-level approach proposed in this article can also be suitable for such systems.