2. 3PRRR: Prototype and Modeling

In this section, the prototype 3PRRR is first described (see Fig. 1). Then, the visual system and image processing technique are introduced. Finally, its kinematic and dynamic models are briefly derived for the sake of completeness. Fontes et al. [Reference Fontes and da Silva15] describe these models in detail. The geometric parameters required for the derivation of the kinematic and dynamic models of the 3PRRR manipulator are illustrated in Fig. 2.

2.1. Prototype

Each kinematic chain presents an active prismatic joint (P) and an active revolute joint (R). The actuation of these joints is performed by brushless Maxon EC60 flat motors delivering 100 W of power. These motors are coupled with Maxon planetary gearheads GP52C with a reduction ratio of 3.5:1, resulting in a nominal torque of 0.82 N.m @ 1200 rpm. Table systems with ball screw HIWIN KK60-10-C-E-600-A-1-F0-S3 are responsible for the linear motion. Their stroke range is 600 mm, and their lead is 10 mm. The Maxon EPOS2 50/5 controllers generate the actuation signals and acquire the encoders’ measurements. These signals are processed in a DSpace 1103 acquisition board at a rate of $100 \ Hz$ . The communication between the Maxon EPOS2 50/5 controllers and the DSPACE board is performed using CAN protocol. The torques of the active revolute joints, $\boldsymbol{\tau}_{\theta}$ , and the active prismatic joints, $\boldsymbol{\tau}_{\delta}$ , are proportional to these actuation signals.

In Fig. 2, the subscript $i = 1, \ldots, 3$ indicates the kinematic chain. In each kinematic chain, there are one active ( $A_{i}$ ) and two passive ( $B_{i}$ and $C_{i}$ ) revolute joints. The angles $\theta_{i}$ and $\beta_{i}$ represent the orientation of the links $A_{i}B_{i}$ and $B_{i}C_{i}$ , respectively. Their lengths are $l_{1}$ and $l_{2}$ , respectively.

Active prismatic joints can modify the position of the point $A_{i}$ . By using these linear actuators, the position of $A_i$ can be modified according to the position $\delta_{i}$ and the orientation $\gamma_i$ ( $\gamma_{i}=\lambda_i-\pi/2$ ) of the linear guide. The angular position of the active revolute joints, $\boldsymbol{\theta}=[\theta_1,\theta_2,\theta_3]$ , are directly acquired by the encoders, while the position of the active prismatic joints, $\boldsymbol{\delta}=[\delta_1,\delta_2,\delta_3]$ , can be derived by using the encoders’ measurements and the lead values.

The distance between the manipulator’s center and the central position of the linear actuators is represented by a. The distance of $C_{i}$ to the centre of the end-effector is h. The geometric and inertia values are given in Table. I. The inertia values have been obtained using a model identification procedure in a similar way that was proposed by [Reference Thanh, Kotlarski, Heimann and Ortmaier33]. The following inertia data are used in the dynamic modeling: the mass at the actuator $A_i$ is $m_0$ , the link $A_iB_i$ ’s mass and moment of inertia around pivot point $A_i$ are $m_1$ and $I_1$ , the link $B_iC_i$ ’s mass and moment of inertia around pivot point $B_i$ are $m_2$ and $I_2$ and the end effector’s mass and moment of inertia around the mass center are $m_n$ and $I_n$ .

Table I.

Inertial and geometric parameters of the 3PRRR manipulator prototype.

Figure 2.

The geometrical parameters of the 3PRRR.

2.2. Visual system and image processing tecnique

The hybrid joint-task CTC strategy requires the measurement of the end-effector’s pose, $\mathbf{X}=[x,y,\alpha]^T$ . This strategy is carried out using a limited frame rate camera and image processing algorithms. In an eye-to-hand configuration, a USB 3.0 oCam-5CRO-U colour camera is placed on top of the manipulator, capturing an image frame that is parallel to the planar workspace. As illustrated in Fig. 3, the red and green markers on the end-effector have a significant color difference compared to the rest of the image. Color segmentation is a simple image processing method that can detect objects of a specific color in the image frame, such as the circular markers used. Most color cameras use the RGB (Red, Green, Blue) color space to represent the colors of the pixels by a combination of Red, Green, and Blue. However, for color filtering of an image, a common approach is the conversion to the HSV (Hue, Saturation and Value) colour space since only the Hue channel is needed to describe the pixel’s color. Saturation represents the amount of the color, while Value depends on the lighting conditions. After selecting a range of pixel values in either RGB or HSV color spaces that best describe the color of the object to be detected, the image can be segmented by applying a mask or filter. A binary image with the detected object is created by selecting only the pixels that belong to the range specified.

Figure 3.

An image frame used to measure the end-effector’s pose based on the detection of the two colored markers and the coordinate transformation between the uv- and xy-axes.

Two binary images representing the red and green circular markers on the end-effector are extracted. The center of the circles is the average position of all white pixels in each binary image. Geometric relationships between these two points and the end-effector can be used to compute its pose in the image coordinates. The end-effector’s center is the average between the center of the green and red markers (see Fig. 3). The angular position is also easily acquired by considering that the markers’ centers form a parallel line with one side of the end-effector. Although the end-effector’s pose is measured using image-based coordinates, it is possible to convert it into position-based ones, such as the manipulator’s reference system. Since the workspace plane is parallel to the image frame, this coordinate transformation only needs a translation of axes, as seen in Fig. 3. To shift the origin between these coordinate systems, we obtained the manipulator’s coordinates origin in the image frame in a calibration test. For an undistorted image, the corresponding size of a pixel at the workspace plane is used as a scaling factor between these coordinate systems. This algorithm can measure the end-effector’s pose at a sampling time of 60 Hz with an image resolution of 640x480 pixels. Accuracy depends on the relation between in pixel the actual measurement. This is assessed during the system calibration. For the used device, the accuracy is 0.2 mm.

2.3. Kinematic modeling

The IKM is used to determine the active joints inputs $\boldsymbol{\Theta} =[\boldsymbol{\theta} \quad \boldsymbol{\delta}]^T= [\theta_{1},\theta_{2},\theta_{3},\delta_{1},\delta_{2},\delta_{3}]^{T}$ that yield a desired end-effector’s pose $\mathbf{X} = [x,y,\alpha]^{T}$ . So, considering that a redundancy resolution scheme will define the values of the redundant actuators’ inputs, $\boldsymbol{\delta}$ , the inverse kinematics of the manipulator can be derived using the geometrical constraint $\left \| \overrightarrow{B_iC_i} \right \| = l_{2}$ , yielding:

(1) \begin{equation}e_{i1} + e_{i2}\cos(\theta_{i}) + e_{i3}\sin(\theta_{i}) = 0,\end{equation}

where $ e_{i1}=-2l_1\rho_{yi}$ , $e_{i2}=-2l_1\rho_{xi}$ and $e_{i3}=\rho^2_{xi}+\rho^2_{yi}+l^2_1-l^2_2$ . The auxiliary variables are defined by $\rho_{xi} = x + h \cos(\alpha+\lambda_{i}) - \delta_i \cos(\gamma_{i}) - a\cos(\lambda_{i})$ and $\rho_{yi} = y + h \sin(\alpha+\lambda_{i}) - \delta_i\sin(\gamma_{i}) - a\sin(\lambda_{i})$ , where $\gamma_i$ is the orientation of the prismatic joint in the kinematic chain i which can be calculated as $\lambda_i -\pi/2$ .

Using the tangent half-angle substitution, $\theta_{i}$ is employed to solve Eq. (1) for $\theta_{i}$ and can be derived as

(2) \begin{equation} \theta_i=2\tan^{-1}\left ( \frac{-e_{i1}\pm\ \sqrt{e^2_{i1}+e^2_{i2}-e^2_{i3}}}{e_{i3}-e_{i2}} \right ) \quad \text{and} \quad \beta_i=\tan^{-1}\left ( \frac{\rho_{yi}-l_1\sin(\theta_i)}{\rho_{xi}-l_i\cos(\theta_i)} \right ).\end{equation}

According to the signal choice in Eq. (2), these manipulator presents eight possible working modes. In this work, the plus sign is selected for all calculations yielding the working mode illustrates in Fig. 1.

One way to determine the Jacobian matrix $\mathbf{J}$ is by taking the time derivative of the constraint relation described by Eq. (1). The matrix form of this derivation can be written as $\mathbf{A}\dot{\mathbf{X}}=\mathbf{B}\dot{\boldsymbol{\Theta}}$ , where the matrices $\mathbf{A}$ and $\mathbf{B}$ can be defined as

(3) \begin{equation} \mathbf{A} = \begin{bmatrix} a_{11}\;\;\; & a_{12}\;\;\; & a_{13}\\ a_{21}\;\;\; & a_{22}\;\;\; & a_{23}\\ a_{31}\;\;\; & a_{32}\;\;\; & a_{33} \end{bmatrix} \textrm{ and} \quad \mathbf{B} = \begin{bmatrix} b_{11}\;\;\; & 0\;\;\; & 0\;\;\; & b_{14}\;\;\; & 0\;\;\; & 0\\ 0\;\;\; & b_{22}\;\;\; & 0\;\;\; & 0\;\;\; & b_{25}\;\;\; & 0\\ 0\;\;\; & 0\;\;\; & b_{33}\;\;\; & 0\;\;\; & 0\;\;\; & b_{36}\;\;\; \end{bmatrix}, \end{equation}

where $ a_{i1}=l_2\cos(\beta_i)$ , $a_{i2}=l_2\sin(\beta_i)$ , $a_{i3}=l_2h\sin(\beta_i-\lambda_i-\alpha)$ , $b_{ii}=l_1l_2\sin(\beta_i-\theta_i)$ and $b_{i i+3}=l_2\cos(\beta_i-\gamma_i)$ . The Jacobian matrix can be defined as $ \mathbf{J}=\mathbf{A}^{-1}\mathbf{B}$ .

2.4. Dynamic modeling

The dynamic model of the manipulator is derived in this section using the Newton–Euler formulation and the previously described kinematics. According to Table I, the subscripts $j=0,1,2$ are related to the actuator $A_i$ , link $A_iB_i$ and link $B_iC_i$ , while n is related to the end-effector.

The relation between the generalized efforts $\boldsymbol{\tau}=[\boldsymbol{\tau}_{\theta} \quad \boldsymbol{\tau}_{\delta}]^T$ applied by the actuators and the forces and moments on the system are expressed with the Principle of Virtual Work [Reference Wu, Wang and You36]. This relation uses the components of the vector $\mathbf{p}_{ij}$ , which are composed by the combination of forces and moment applied on the body j of the chain i; the displacement vector $\mathbf{d}_{ij}=[r_{x_{ij}} \quad r_{y_{ij}} \quad \phi_{ij}]^T$ ; and the components of the vector $\mathbf{p}_{n}$ composed by the forces and moment applied on the end-effector. Since the dynamic model is defined using virtual work, it is necessary to obtain Jacobian matrices $\mathbf{K}_{ij}$ for each moving body of the manipulator. The detailed definition for these matrices can be found in [Reference Fontes and da Silva15]. This strategy yields the following relation:

(4) \begin{equation} \boldsymbol{\tau}=\mathbf{J}^T\mathbf{p_n} +\sum_{i=1}^{3}\sum_{j=0}^{2}\mathbf{K_{ij}}^T\mathbf{p_{ij}}.\end{equation}

The vectors $\mathbf{p_{ij}}$ and $\mathbf{p_n}$ can be described:

(5) \begin{equation}\mathbf{p_{ij}}=\mathbf{Z_{ij}} \begin{bmatrix} \ddot{r}_{x_{ij}}\\[3pt] \ddot{r}_{y_{ij}}\\[3pt] \ddot{\phi}_{ij} \end{bmatrix}+ \mathbf{N_{ij}} \begin{bmatrix} \dot{r}_{x_{ij}}\\[3pt] \dot{r}_{y_{ij}}\\[3pt] \dot{\phi}_{ij} \end{bmatrix} \quad \text{and} \quad \mathbf{p_n}=\mathbf{Z_n}\begin{bmatrix} \ddot{x}\\[3pt] \ddot{y}\\[3pt] \ddot{\alpha} \end{bmatrix}\end{equation}

where

(6) \begin{equation}\mathbf{Z_{ij}}=\begin{bmatrix} m_j & 0 & \;\;-m_j s_{ij}\sin{\phi_{ij}} \\ 0 & m_j & \;\;m_j s_{ij}\cos{\phi_{ij}} \\ -m_j s_{ij}\sin{\phi_{ij}}\;\;\;\; &m_j s_{ij}\cos{\phi_{ij}}\;\;\;\; & I_j \end{bmatrix},\end{equation}

(7) \begin{equation}\mathbf{N_{ij}}=\begin{bmatrix} 0\;\;\; & 0\;\;\; & -m_j \dot{\phi}_{ij} s_{ij}\cos{\phi}_{ij} \\[3pt] 0\;\;\; & 0\;\;\; & -m_j \dot{\phi}_{ij} s_{ij}\sin{\phi}_{ij} \\[3pt] 0\;\;\; & 0\;\;\; & 0\end{bmatrix} \quad \text{and} \quad \mathbf{Z_{n}}=\begin{bmatrix} m_n\;\;\; & 0\;\;\; & 0 \\ 0\;\;\; & m_n\;\;\; & 0 \\ 0\;\;\; & 0\;\;\; & I_n \end{bmatrix}\end{equation}

These equations can be rewritten in term of the time derivatives of $\boldsymbol{\Theta}$ using the Jacobian matrices yielding:

(8) \begin{equation} [\tau_1 \quad \tau_2 \quad \tau_3 \quad \tau_4 \quad \tau_5 \quad \tau_6 ]^T= \mathbf{M}\ddot{\boldsymbol{\Theta}}+\mathbf{V}\dot{\boldsymbol{\Theta}},\end{equation}

where:

(9) \begin{equation} \mathbf{M}=(\mathbf{J}^T\mathbf{Z_n}\textbf{J} +\sum_{i=1}^{3}\sum_{j=0}^{2}\mathbf{K_{ij}}^T\mathbf{Z_{ij}}\mathbf{K_{ij}}) \quad \text{and}\end{equation}

(10) \begin{equation} \mathbf{V}=(\mathbf{J}^t\mathbf{Z_n}\mathbf{{J}} +\sum_{i=1}^{3}\sum_{j=0}^{2}\mathbf{K_{ij}}^T\mathbf{Z_{ij}}\mathbf{{K}_{ij}}+\sum_{i=1}^{3}\sum_{j=0}^{2}\mathbf{K_{ij}}^t\mathbf{N_{ij}}\mathbf{K_{ij}}).\end{equation}

Since $\tau_4$ , $\tau_5$ , and $\tau_6$ are generalized forces, they must be converted to torque using the lead K of the linear guide leading to

(11) \begin{equation} \boldsymbol{\tau}= [\boldsymbol{\tau}_{\theta} \quad \boldsymbol{\tau}_{\delta}]^T = [\tau_1 \quad \tau_2 \quad \tau_3 \quad K\tau_4/(2\pi) \quad K\tau_5/(2\pi) \quad K\tau_6/(2\pi)]^T\end{equation}

It is important to highlight that the dynamic model depends on the angles $\phi_{ij}$ . These angles are defined as 0 for $j=0$ , $\theta_i$ for $j=1$ and $\beta_i$ for $j=2$ . This dependency is critical for the control strategies based on the model, since $\beta_i$ can only be analytically calculated by the IKM.

3. Redundancy Resolution

Redundancy resolution is a decision-maker strategy used for deriving the position of the redundant joints among the possibilities. This strategy can be mathematically described as an optimization problem. Some redundancy resolution schemes seek to minimize the maximum required torque to perform a predefined task. However, this approach does not guarantee the avoidance of singular regions during the task execution.

In this work, a multiobjective optimization problem is adopted to overcome this issue. In the proposed approach, the minimization of the required torque is considered and the avoidance of the singular regions. In this way, two functions are defined, $\upsilon_1$ and $\upsilon_2$ whose values present a strong relation with the distance from the type 1 and type 2 singularities, respectively. According to Pagis et al., type 1 singularities occur when the mechanism is unable to move in a given direction due to the rank deficiency of the kinematic matrix $\mathbf{B}$ (Eq. (3)), and type 2 singularities occur when one or more of the manipulator’s degrees of freedom become uncontrollable due to the rank deficiency of the kinematic matrix $\mathbf{B}$ (Eq. (3)). In this way, penalty functions are proposed for guaranteeing the avoidance of types 1 and 2 singularities.

First, a function $\upsilon_1$ is proposed for the avoidance of type 1 singularities. These singularities can be avoided by assuring that the values of $\sin(\beta_i-\theta_i)$ , from Eq. (12), are positive. Equation 12 only considers the nonredundant part of the matrix ${\mathbf{B}}$ (see Eq. (3)) in the following way:

(12) \begin{equation} \mathbf{B}' = \begin{bmatrix} l_1l_2\sin(\beta_1-\theta_1)\;\;\; & 0 & 0\\ 0 & l_1l_2\sin(\beta_2-\theta_2)\;\;\; & 0\\ 0 & 0 & l_1l_2\sin(\beta_3-\theta_3) \;\;\; \end{bmatrix}.\end{equation}

Considering that the singular values of $\mathbf{B}'$ are related to $\sin(\beta_i-\theta_i)$ , the avoidance of type 1 singularity can be guaranteed by maximizing the smallest singular value of $\mathbf{B}'$ . Since the smallest singular value varies in time during the task execution, the minimum value of the smallest singular values (worst case) is considered in the optimization problem. In this way, the function $\upsilon_1$ is defined by

(13) \begin{equation} \upsilon_1 = min \quad \frac {min \ \nu(\mathbf{B}')}{l_1l_2},\end{equation}

where $\nu(\mathbf{B}')$ are the singular values of matrix $\mathbf{B}'$ , and $l_1$ and $l_2$ are used for normalizing the relation. The value of $\upsilon_1$ varies between 0 and 1 and should be maximized for guaranteeing the avoidance of type 1 singularity.

Second, a function $\upsilon_2$ is proposed for the avoidance of type 2 singularities. This requirement can be assessed by evaluating indices derived from the Jacobian matrix $\mathbf{A}$ [Reference Alba-Gomez, Wenger and Pamanes37]. For the manipulator under study, the Jacobian matrix $\mathbf{A}$ (see Eq. (3)) is not homogeneous and, therefore, the indices defined directly from it have no physical meaning. To overcome this problem, the Jacobian matrix $\mathbf{A}$ could be normalized using the characteristic length of the manipulator, given by $Lc = \sqrt{2}h $ [Reference Alba-Gomez, Wenger and Pamanes37,Reference Reveles, Pamanes and Wenger38]. This normalization can be applied to the angular-related elements of the matrix $\mathbf{A}$ yielding the normalized matrix:

(14) \begin{equation} \mathbf{{A}}' = \begin{bmatrix} a_{11}\;\;\; & a_{12}\;\;\; & a_{13}/Lc\\ a_{21}\;\;\; & a_{22}\;\;\; & a_{23}/Lc\\ a_{31}\;\;\; & a_{32}\;\;\; & a_{33}/Lc \end{bmatrix}.\end{equation}

The distance between the end-effector and the singularities of type 2 is related to the conditioning number of the normalized Jacobian matrix ${\mathbf{A}}'$ defined by

(15) \begin{equation} \kappa({\mathbf{A}}')=\frac{max\ \nu({\mathbf{A}}')}{min\ \nu({\mathbf{A}}')}.\end{equation}

where $\nu({\mathbf{A}}')$ are the singular values of the matrix ${\mathbf{A}}'$ . Since $\kappa({\mathbf{A}}')$ varies in time during the task execution, its maximum value is going to be considered in the cost function. In this way, the function $\upsilon_2$ is defined by

(16) \begin{equation} \upsilon_2 = max \quad \kappa({\mathbf{A}}').\end{equation}

The value of $\nu({\mathbf{A}}')$ varies from 1 to $\infty$ . The higher the value of $\nu({\mathbf{A}}')$ , the nearer to a singularity the end-effect is. In this way, the avoidance of type 2 singularity can be guaranteed by minimizing the $\upsilon_2$ .

The proposed multiobjective optimization problem seeks to minimize the maximum required torque ( $\left \| \boldsymbol{\tau} \right \|_\infty$ ), to maximize the function $\upsilon_1$ for avoiding type 1 singularities and to minimize the function $\upsilon_2$ for avoiding type 2 singularities. A weighted sum strategy is proposed to convert this multiobjective problem into a single optimization problem defined by

(17) \begin{equation} F_{cost} = \left \| \boldsymbol{\tau} \right \|_\infty + w[\upsilon_2 - \upsilon_1],\end{equation}

where w is a weighting factor. The definition of this objective function yields the following nonlinear constrained optimization problem:

(18) \begin{equation} \begin{split} \delta_{i,opt}(t) = arg &\quad min_{\ \delta_i(t)} \ F_{cost} \\ subject \ to\;:\;\ \delta_{min}&\leq\delta_i(t)\leq\delta_{max}\\ \upsilon_1 &\geq R_1 \\ \upsilon_2 &\leq R_2\end{split}\end{equation}

where $R_1$ and $R_2$ are the lower and upper bounds selected to keep the end-effect far enough from the singularities.

The variables of this optimization problem should describe the inputs for the redundant actuators $\delta_i(t)$ . Since the inputs $\delta_i(t)$ are time-dependent variables, they cannot be directly used as optimization variables. Thus, two different approaches are explored to represent these variables: prepositioning and ongoing positioning approaches [Reference Kotlarski, Abdellatif, Ortmaier and Heimann10].

On the one hand, the prepositioning approach consists of determining the optimal inputs for the redundant active prismatic joints before the execution of the task (the predefined trajectory). In this way, the positions of the active prismatic joints (the linear guides), denoted as $\delta_ {fixed_i,opt}$ , are the same throughout the execution of the task. Therefore, this optimization problem can be described as

(19) \begin{equation} \begin{split} \delta_{fixed_i,opt} = arg &\quad min_{\ \delta_{fixed_i}} \ F_{cost} \\ subject \ to\;:\;\ \delta_{min}&\leq\delta_{fixed_i}\leq\delta_{max}\\ \upsilon_1 &\geq R_1 \\ \upsilon_2 &\leq R_2 \end{split}\end{equation}

On the other hand, the ongoing positioning approach consists of determining the optimal inputs for the redundant active joints considering that they can also move during the execution of the task (the predefined trajectory). A fifth-order polynomial for describing these inputs have been selected. For deriving the coefficients of this polynomial, the initial and final positions, velocities, and accelerations should be known. The initial and final velocities and accelerations are equal to zero. So, the optimization variables are the values of the initial ( $\delta_{0_i,opt}$ ) and final ( $ \delta_{f_i,opt}$ ) positions of the active prismatic joints (the linear guides). Therefore, this optimization problem can be described as

(20) \begin{equation} \begin{split} [{\delta_{0_i,opt},\delta_{f_i,opt}}] = arg &\quad min_{\ [\delta_{0_i},\delta_{f_i}]} \ F_{cost} \\ subject \ to\;:\;\delta_{min}&\leq\delta_{0_i}\leq\delta_{max} \\ \delta_{min}&\leq\delta_{f_i}\leq\delta_{max} \\ \delta_{min}&\leq\delta_{i}\leq\delta_{max} \\ \upsilon_1 &\geq R_1 \\ \upsilon_2 &\leq R_2\end{split}\end{equation}

Both redundancy resolution schemes are performed offline, which means that they must be executed before the experiment. They are solved by using a hybrid optimization strategy. A classical genetic algorithm (GA) is used for delivering good initial guesses for Sequential Quadratic Programming (SQP), which is responsible for providing refined results. This strategy has been able to provide satisfactory results on reasonable processing time. The outcome of this optimization is the desired positions for the active redundant actuators (the active prismatic joints) which is denoted as $\boldsymbol{\delta}_d(t)$ . The values of $\boldsymbol{\delta}_d(t)$ are used as the references for the control strategies.

4. Control Strategies

For a predefined trajectory $\mathbf{X}_d$ , a set of desired positions for the active redundant joints $\boldsymbol{\delta}_d$ can be derived by one of the redundancy resolution schemes (Eqs. (19) or (20)). The desired positions of the active nonredundant joints $\boldsymbol{\theta}_d$ can be derived by using the IKM, described in Section 2.3, for given values of $\mathbf{X}_d$ and $\boldsymbol{\delta}_d$ . For a given $\boldsymbol{\Theta}_d =[\boldsymbol{\theta}_d \quad \boldsymbol{\delta}_d]^T$ , the required torque to be applied in the manipulator is derived using three control strategies in this work. These strategies are experimentally implemented and compared.

The first strategy is a model-free independent PID joint control scheme. This strategy is illustrated in Fig. 4. The desired positions of the active joints, $\boldsymbol{\theta}_d$ and $\boldsymbol{\delta}_d$ , are used as reference signals in the control loop. These signals are compared with the actual measurement, $\boldsymbol{\theta}$ and $\boldsymbol{\delta}$ , and the differences between these signals, $\boldsymbol{e}_{\theta}=\boldsymbol{\theta}_d-\boldsymbol{\theta}$ and $\boldsymbol{e}_{\delta}=\boldsymbol{\delta}_d-\boldsymbol{\delta}$ , are the inputs for the PID controls. Different gains are used for the actuators responsible for the motion of the active revolute (nonredundant) and prismatic (redundant) joints. These gains have been experimentally adjusted by a trial-and-error method. The torque calculated using this strategy, $\boldsymbol{\tau}_{PID}$ , can be calculated using the following control law:

Figure 4.

The independent PID joint control scheme (PID).

(21) \begin{equation} \boldsymbol{\tau}_{PID} = [\boldsymbol{\tau}_{\theta}, \ \boldsymbol{\tau}_{\delta}]=K_p\mathrm{e}+K_d\frac{\mathrm{d} e}{\mathrm{d} t}+K_i\int_{0}^{t}e \ \mathrm{d}\rho\end{equation}

where $\boldsymbol{e} = [\boldsymbol{e}_{\theta},\boldsymbol{e}_{\delta}$ ], $\boldsymbol{K}_p$ , $\boldsymbol{K}_d$ and $\boldsymbol{K}_i$ are the proportional, derivative and integral gains.

The second strategy is a joint space CTC scheme. Figure 5 shows a scheme of the second strategy. In this strategy, a feedforward signal is added since an inverse dynamic model (IDM), derived in Section 2.4, is available. Based on the IKM, the desired joint positions $[\boldsymbol{\theta}_d, \boldsymbol{\delta}_d]$ , velocities $[\dot{\boldsymbol{\theta}}_d, \dot{\boldsymbol{\delta}}_d]$ and accelerations $[\ddot{\boldsymbol{\theta}}_d, \ddot{\boldsymbol{\delta}}_d]$ are derived. Moreover, the desired angular positions of the passive joints $\boldsymbol{\beta}_d$ are also derived using Eq. (2). These desired values are used in the Eq. (4) to compute the required torque to the task execution (the feedforward signal):

Figure 5.

The CTC strategy.

(22) \begin{equation} \boldsymbol{\tau}_{ff}=[\boldsymbol{\mu}_{ff\theta}, \ \boldsymbol{\mu}_{ff\delta}] =\text{IDM}(\boldsymbol{\beta}_d,\boldsymbol{\theta}_d,\dot{\boldsymbol{\theta}}_d,\ddot{\boldsymbol{\theta}}_d,\boldsymbol{\delta}_d,\dot{\boldsymbol{\delta}}_d,\ddot{\boldsymbol{\delta}}_d)\end{equation}

This computed torque (feedforward signal) may enhance the system performance regarding the tracking reference and disturbance rejection capabilities since it might compensate for Coriolis, centrifugal, frictional forces, among others. Nevertheless, the PID scheme is also used in this strategy for robustness since the dynamic model is surely imprecise. This CTC strategy yields the following control law:

(23) \begin{equation} \boldsymbol{\tau}_{CTC} = [\boldsymbol{\tau}_{\theta}, \ \boldsymbol{\tau}_{\delta}]=[\boldsymbol{\mu}_{ff\theta}+\boldsymbol{\mu}_{\theta}, \ \boldsymbol{\mu}_{ff\delta}+\boldsymbol{\mu}_{\delta}]\end{equation}

where $\boldsymbol{\mu}_{\theta}$ and $\boldsymbol{\mu}_{\delta}$ are the output of the PID controllers as can be seen in Fig. 5.

The third strategy is a hybrid joint-task space CTC strategy. Figure 6 shows a scheme of the third strategy. While the computed torque is derived using the desired kinematic quantities (Eq. (22)) in the second control strategy, the actual measurements will be used in the control loops in the third control strategy. Not only the actual positions of the nonredundant and redundant actuators, $\boldsymbol{\delta}$ and $\boldsymbol{\theta}$ (joint space), but also the actual positions of the end-effector, $\boldsymbol{X}$ (task space), are used. The task space is measured using a camera and image processing algorithms (see Section 2.1).

Figure 6.

The Hybrid Joint-Task Space computed torque control strategy (HYB).

The hybrid joint-task space CTC strategy, illustrated in Fig. 6, presents two control loops: an internal loop that considers the dynamic model and an external loop that tracks the error. For these loops, kinematic quantities are required. The first- and second-time derivatives of the actual actuators’ positions are calculated using derivative blocks and estimated signals. The estimated second-time derivative of the nonredundant actuators’ positions are calculated by

(24) \begin{equation} \ddot{\boldsymbol{\theta}}_*=\ddot{\boldsymbol{\theta}}_d+\boldsymbol{u}_{\theta}\end{equation}

where $\boldsymbol{u}_{\theta}$ is the outcome of PID blocks while the vector of errors of the actuators’ positions ( $\boldsymbol{e}_{\theta}=\boldsymbol{\theta}_d-\boldsymbol{\theta}$ ) is the input. The same approach is used for estimating the second-time derivative of the redundant actuators’ positions, $\ddot{\boldsymbol{\delta}}_*$ .

The computed torque is calculated using the IDM, which requires not only the actuators’ positions but also the angle of the passive joints $\boldsymbol{\beta}$ . Using the IKM, the actual angular positions of the passive joints $\boldsymbol{\beta}$ can be also estimated using the actual positions of the active joints, $\boldsymbol{\theta}$ , and of the end-effector, $\boldsymbol{X}$ (see Eq. (2)). In this way, these states are used as input signals along with the kinematic quantities of the actuators. In summary, the required torque for executing a task is derived using not only the IKM and the IDM but also the actual and estimated kinematic quantities:

(25) \begin{equation} \boldsymbol{\tau}_{HYB} = [\boldsymbol{\tau}_{\theta}, \ \boldsymbol{\tau}_{\delta}] = \text{IDM}(\boldsymbol{\beta},\boldsymbol{\theta},\dot{\boldsymbol{\theta}},\ddot{\boldsymbol{\theta}}_*,\boldsymbol{\delta},\dot{\boldsymbol{\delta}},\ddot{\boldsymbol{\delta}}_*)\end{equation}

Both second and third strategies require the calculation of the inertia matrix and the Coriolis vector. These three control strategies are implemented in a Simulink model, compiled for a dSPACE 1103 board. The strategies use the same PID gains for the sake of comparison. In the next section, these strategies are compared considering the RMS (root mean square) of the error vectors ( $\boldsymbol{e}_{\theta}=\boldsymbol{\theta}_d-\boldsymbol{\theta}$ and $\boldsymbol{e}_{\delta}=\boldsymbol{\delta}_d-\boldsymbol{\delta}$ ) and the RMS of the torques ( $\boldsymbol{\tau}_{\theta}$ and $\boldsymbol{\tau}_{\delta}$ ).