**3. Kinematic model of the cable shovel front end**

A kinematic model of the shovel is required to completely describe the motions (accelerations, velocities, and displacements). The kinematic model further provides a basis for the dynamic model, which can be used to calculate the torques and forces on individual components. The complete shovel digging process involves propel, crowd, and swing motions. However, during the normal duty cycle, the shovel positions itself against the working face without propel. In this situation, only the front-end assembly moves. Further, the maximum forces are involved during the excavation phase. Therefore, a dynamic model of the front-end assembly alone can suffice to describe the normal duty cycle of cable shovel. **Figure 3** shows the shovel front-end assembly, whose mechanism is modeled as a three-link system (saddle, crowd arm, and dipper) with three links and three joints. The saddle is a fixed length link and is free to rotate in the vertical plane. The rotation of the saddle block controls the vertical position of the dipper. The crowd arm is connected to the saddle block through a prismatic joint, and its length varies during the crowding action of the digging operation.

The length of the crowd arm controls the horizontal position of the dipper. The crowd arm and the saddle have the same rotation cycle, while the dipper is oriented at a fixed angle, β, to the crowd arm. The dipper is also a fixed length link. The rotation of the saddle block and the length of the crowd arm together control the position of the dipper in the vertical plane and its trajectory. The structural kinematic parameters of the shovel using the Denavit-Hartenberg (D-H) notation [12] are represented in **Figure 3** and **Table 3**. Here, four values are assigned to each link following the D-H notation.

**Figure 3.** Structural kinematics using D-H procedure.

The forces, f1 and f5, are the dynamic forces [7], where f1 changes both in magnitude and the point of application, and f5 depends on the bucket acceleration. The force (f6) was originally defined as a part of f1 and f5. The dipper payload force (f1) is the dominant force for the largecapacity dippers [6, 8, 9]. Awuah-Offei et al. [8] proposed a model based on the Balovnev [10] excavation model using the six forces. The force (f2) can be set to zero [6]. Forces, f3 and f4, are the cutting forces and can be combined as a single force and estimated using the empirical

This empirical model is a result of extensive experimentation on frozen soils [11]. z is the coefficient that accounts for the blade impact on cutting force, which depends on w and d. **Table 1** is used to estimate z for d (between 25 cm and 50 cm). z increases as d decreases, and it also depends on the ratio Ts/Tw (Ts is the spacing between the teeth, and Tw is the tooth width). **Table 2** lists the multiplying factors for z based on Ts/Tw. Force f5 can also be set to zero if the dipper moves with a uniform velocity through the muck pile. Force f6 can be modeled as part

Length of horizontal surface (w, m) 0.25–0.50 0.50–0.75 0.75–1.00 1.00–1.25 Coefficient z 0.55–0.75 0.63–0.78 0.69–0.8 0.71–0.82

Ratio Ts/Tw Ts = Tw Ts = 2Tw−3Tw Ts = 4Tw Ts = 5Tw z 1.2 1 1.1 1.25

A kinematic model of the shovel is required to completely describe the motions (accelerations, velocities, and displacements). The kinematic model further provides a basis for the dynamic model, which can be used to calculate the torques and forces on individual components. The complete shovel digging process involves propel, crowd, and swing motions. However, during the normal duty cycle, the shovel positions itself against the working face without propel. In this situation, only the front-end assembly moves. Further, the maximum forces are involved during the excavation phase. Therefore, a dynamic model of the front-end assembly alone can suffice to describe the normal duty cycle of cable shovel. **Figure 3** shows the shovel front-end assembly, whose mechanism is modeled as a three-link system (saddle, crowd arm, and dipper) with three links and three joints. The saddle is a fixed length link and is free to rotate in the vertical plane. The rotation of the saddle block controls the vertical position of the dipper. The

**3. Kinematic model of the cable shovel front end**

1.35 P=10C d (1+2.6w)(1+0.0075 ')z <sup>o</sup> b (1)

model [11] given by Eq. (1).

110 Lagrangian Mechanics

**Table 1.** Dependence of "z" on "d," and "w".

**Table 2.** Dependence of z on Ts/Tw [11].

of f1:


**Table 3.** Structural kinematic parameters.

The two values (ai, di) are for the links and represent the constant and variable lengths of the links, while the other two (αi, θi) are for the connection between links (i.e., joints), and, thus, represent the rotation of the coordinate frame and rotation of the joint, respectively. For a revolute joint, ai, αi, and di are fixed and θi is a variable. On the other hand, for a prismatic joint (or translational motion), ai, αi, and θi are fixed and di is a variable. The crowd-arm movement is via a prismatic joint. A kinematic scheme relates the movements of the links and translates the motions and rotations in the reference coordinate frame. The D-H scheme is used to relate the movements and rotation of the links. The movements and rotations of individual links are measured in the coordinate frames assigned at every joint location using the D-H scheme [12]. The lower part of the shovel is stationary and fixed for this analysis.

The XoYoZo frame, the reference coordinate frame, is selected with Zo along the rotating axis of the saddle block. The coordinate frame X1Y1Z1 coincides with the XoYoZo frame and measures the rotation of the dipper handle via the saddle block. Next, the coordinate frame X2Y2Z2 is set at the intersection of the saddle block and the dipper handle, with the Z2 axis along the translation movement of the dipper handle (joint 2 being a prismatic joint). The movement of the dipper handle is measured along this Z2 axis. The coordinate frame X3Y3Z3 is set at the end point of the dipper handle with Z3 normal to Z2. This frame is at a fixed angle from coordinate frame 2. And finally, the frame X4Y4Z4 is set at the tip of the dipper with Z4-axis parallel to Z3. The material resistive forces acting on the shovel are defined in this frame. The coordinate frame assignments are also shown in **Figure 3**.

*Forward kinematics of cable shovel front*-*end assembly*: The forward kinematic model defines the positions and motions of the dipper with known dipper-handle rotation and extension. External dynamic forces act on the shovel dipper during excavation. A transformation scheme is used to translate point coordinates in one coordinate frame to the first coordinate frame. The homogenous transformation matrix for transferring coordinates from *i*−*1* coordinate frame to *i* frame, in its general form for revolute and prismatic joints is given in Eqs. (2) and (3), respectively [13]. These equations can be derived considering two links (*i*−*1* and *i*) connected through revolute or prismatic joints, respectively. These transformation equations are a combination of rotation and translation matrices:

$$\begin{aligned} T\_l^{i-1} &= \begin{bmatrix} \cos \theta\_l & -\cos \alpha\_l \mathbf{a}\_l \sin \theta\_l & \sin \alpha\_l \sin \theta\_l & \mathbf{a}\_l \cos \theta\_l \\ \sin \theta\_l & \cos \alpha\_l \cos \theta\_l & -\sin \alpha\_l \cos \theta\_l & \mathbf{a}\_l \sin \theta\_l \\ 0 & \sin \alpha\_l & \cos \alpha\_l & \mathbf{d}\_l \\ 0 & 0 & 0 & 1 \end{bmatrix} \\\\ T\_l^{i-1} &= \begin{bmatrix} \cos \theta\_l & -\cos \alpha\_l \mathbf{a}\_l \sin \theta\_l & \sin \alpha\_l \sin \theta\_l & 0 \\ \sin \theta\_l & \cos \alpha\_l \cos \theta\_l & -\sin \alpha\_l \cos \theta\_l & 0 \\ 0 & \sin \alpha\_l & \cos \alpha\_l & \mathbf{d}\_l \end{bmatrix} \end{aligned} \tag{2}$$

The individual transformation matrices Ti i−1 are formulated using Eqs. (2) and (3). These matrices relate the geometry of a point in the two adjacent coordinate frames as in **Figure 3** and can further be multiplied together to obtain a transformation matrix between any two coordinate frames. These transformations are required for the shovel front-end kinematic and dynamic models using the Newton-Euler procedure. The Newton-Euler method is an iterative method for computing the velocities, accelerations, joint torques, and forces from crowd arm

to dipper in the forward direction and from dipper tip to the saddle block in the reverse direction. Newton-Euler method has an advantage of being iterative, which makes it more suitable for computer simulations.

revolute joint, ai, αi, and di are fixed and θi is a variable. On the other hand, for a prismatic joint (or translational motion), ai, αi, and θi are fixed and di is a variable. The crowd-arm movement is via a prismatic joint. A kinematic scheme relates the movements of the links and translates the motions and rotations in the reference coordinate frame. The D-H scheme is used to relate the movements and rotation of the links. The movements and rotations of individual links are measured in the coordinate frames assigned at every joint location using the D-H

The XoYoZo frame, the reference coordinate frame, is selected with Zo along the rotating axis of the saddle block. The coordinate frame X1Y1Z1 coincides with the XoYoZo frame and measures the rotation of the dipper handle via the saddle block. Next, the coordinate frame X2Y2Z2 is set at the intersection of the saddle block and the dipper handle, with the Z2 axis along the translation movement of the dipper handle (joint 2 being a prismatic joint). The movement of the dipper handle is measured along this Z2 axis. The coordinate frame X3Y3Z3 is set at the end point of the dipper handle with Z3 normal to Z2. This frame is at a fixed angle from coordinate frame 2. And finally, the frame X4Y4Z4 is set at the tip of the dipper with Z4-axis parallel to Z3. The material resistive forces acting on the shovel are defined in this frame. The coordinate

*Forward kinematics of cable shovel front*-*end assembly*: The forward kinematic model defines the positions and motions of the dipper with known dipper-handle rotation and extension. External dynamic forces act on the shovel dipper during excavation. A transformation scheme is used to translate point coordinates in one coordinate frame to the first coordinate frame. The homogenous transformation matrix for transferring coordinates from *i*−*1* coordinate frame to *i* frame, in its general form for revolute and prismatic joints is given in Eqs. (2) and (3), respectively [13]. These equations can be derived considering two links (*i*−*1* and *i*) connected through revolute or prismatic joints, respectively. These transformation equations are a

matrices relate the geometry of a point in the two adjacent coordinate frames as in **Figure 3** and can further be multiplied together to obtain a transformation matrix between any two coordinate frames. These transformations are required for the shovel front-end kinematic and dynamic models using the Newton-Euler procedure. The Newton-Euler method is an iterative method for computing the velocities, accelerations, joint torques, and forces from crowd arm

(2)

(3)

i−1 are formulated using Eqs. (2) and (3). These

scheme [12]. The lower part of the shovel is stationary and fixed for this analysis.

frame assignments are also shown in **Figure 3**.

112 Lagrangian Mechanics

combination of rotation and translation matrices:

The individual transformation matrices Ti

The propagation of angular and linear velocities from joint to joint is given by Eqs. (4) through (7) [13]. For rotational motion, the angular and linear velocities are defined by Eqs. (4) and (5), respectively. For prismatic joint, the angular and linear velocity relations are given by Eqs. (6) and (7), respectively:

$$\mathbf{R}^{i+1}\mathbf{a}\mathbf{o}\_{i+1} = \mathbf{a}^{i+1}\mathbf{R}^{\ i}\mathbf{a}\mathbf{o}\_{i} + \hat{\mathbf{o}}\_{i+1} \quad \text{ ${}^{i+1}\hat{Z}\_{i+1}$ }\tag{4}$$

$$\mathbf{v}^{i+1}\mathbf{v}\_{i+1} = \mathbf{v}^{i+1}\_{\;i}\mathbf{R}\left(^{i}\mathbf{v}\_{i} + ^{i}\mathbf{o}\_{i}\mathbf{X}\;^{i}\mathbf{P}\_{i+1}\right) \tag{5}$$

$$\mathbf{R}^{i+1}\mathbf{\dot{o}}\_{i+1} = \mathbf{\dot{s}}^{i+1}\mathbf{R}^{i}\mathbf{\dot{o}}\_{i} \tag{6}$$

$$\mathbf{v}^{\;\;\!\perp \!+1} \mathbf{v}\_{\vert \!\perp \!+\!} = \, ^{\!\perp \!+\!\!\perp} \mathbf{R} \left( ^{\!\perp} \mathbf{v}\_{\vert} + ^{\!\perp} \mathbf{o}\_{\vert} \, \mathbf{X} \, ^{\!\perp} \mathbf{P}\_{\vert \!\perp \!+\!\!\perp} \right) + \dot{\mathbf{d}}\_{\vert} \hat{\mathbf{Z}}\_{\vert} \tag{7}$$

The required rotation matrices are derived from the transformation matrices in Eqs. (2) and (3). The 3x3 matrix, within a transformation matrix Ti i−1, represents the corresponding rotation matrix Ri i−1. The forward kinematic starts from the first link (saddle block) and moves outward toward the last link (dipper). The objective is to determine the propagation of the joint rotation and velocities from the joint 1 to the dipper tip. The model uses the same start point equations and basic simplifying assumption from Frimpong et al. [14], and as a result, the kinematic equations are very similar as well. However, the resulting dynamic model is different due to the improved resistive forces in this model. The reference frame {0} is fixed with the lower frame through the boom. The lower structure of the shovel is fixed, so its linear and angular velocities and accelerations remain zero at all times during the excavation as shown in Eqs. (8) and (10). These values change only during the propel motion of shovel which is not considered in this research. The joint velocity can be determined by taking the derivative of rotation of joint 1 as shown in Eqs. (9) and (11), respectively. Similarly, the linear velocity of the stationary lower structure of the shovel is zero:

$$\mathbf{a}^{\circ}\boldsymbol{\alpha}\_{0} = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} \tag{8}$$

$$\frac{d\,^0 \alpha\_0}{dt} = \,^0 \dot{\alpha}\_0 = \begin{bmatrix} 0\\0\\0\\0 \end{bmatrix} \tag{9}$$

$$\mathbf{v}^{\mathbf{0}}\mathbf{v}\_{0} = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} \tag{10}$$

$$\frac{d^0 \mathbf{v}\_0}{dt} = \,^0 \dot{\mathbf{v}}\_0 = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} \tag{11}$$

Eq. (12) is obtained from Eq. (4) for joint 1 (*i* = 0), a revolute joint. It is evident from this equation that the angular velocity of the first link is only around Z-axis and is equivalent to the rate of change of angular rotation around joint 1. The linear velocity propagation to joint 1 can be computed using Eq. (5) as Eq. (13). The first link experiences only the rotational motion. Therefore, the linear velocity of joint 1 is zero:

$$\begin{aligned} \,^1\boldsymbol{\alpha}\_1 &= \,^1\_0 \boldsymbol{R}^0 \,^0 \boldsymbol{\alpha}\_0 + \dot{\boldsymbol{\theta}}\_1 \,^1 \hat{\mathbf{Z}}\_1 = \boldsymbol{\alpha}\_1 = \,^1\_0 \boldsymbol{R}^0 \,^0 \boldsymbol{\alpha}\_0 + \dot{\boldsymbol{\theta}}\_1 \,^1 \hat{\mathbf{Z}}\_1 = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \\ \dot{\boldsymbol{\theta}}\_1 \end{bmatrix} \end{aligned} \tag{12}$$

$$\mathbf{v}^{\prime}\mathbf{v}\_{\parallel} = \,\_{0}^{1}\mathrm{R}\left(\,^{\diamond}\mathbf{v}\_{\phi} + \,^{\diamond}\mathrm{o}\mathbf{o}\_{0}\,\mathrm{X}\,^{\diamond}\mathrm{P}\_{\parallel}\right) = \begin{bmatrix} 0\\0\\0 \end{bmatrix} \tag{13}$$

For the prismatic joint 2 (*i* = 1), Eq. (6) computes the angular velocity of the link 2 (the crowd arm). The propagation of angular velocity to joint 2, as given in Eq. (14), shows that the angular velocity of joint 2 is dependent upon the rate of change of angular rotation of joint 1, and there is only an axis shift involved (from Z-axis to Y-axis) during the propagation:

$$\begin{aligned} \mathbf{^2o\_2} &= \mathbf{^2R}\,^1o\_1 + \dot{\theta}\_2\,^2\hat{Z}\_2 = \begin{bmatrix} \mathbf{0} \\ \dot{\theta}\_1 \\ \mathbf{0} \end{bmatrix} \\ \text{ (14) } &\ddot{\theta}\_2\,\text{, and }\ \ddot{\theta}\_2\,\text{ are zero} \end{aligned} \tag{14}$$

Mechanics of Electric Rope Shovel Performance and Reliability in Formation Excavation http://dx.doi.org/10.5772/65333 115

$$\mathbf{v}^{2}\mathbf{v}\_{2} = \,^{2}\_{1}\mathbf{R}\left(^{1}\mathbf{v}\_{1} + ^{1}\boldsymbol{\alpha}\_{1}\mathbf{X}^{-1}\mathbf{P}\_{2}\right) = \begin{bmatrix} 0\\0\\-\mathbf{a}\_{1}\dot{\boldsymbol{\theta}}\_{1} \end{bmatrix} \tag{15}$$

The linear velocity propagation to joint 2 is calculated as Eq. (15). Similarly, the angular and translational velocities are calculated for joint 3 as Eqs. (16) and (17). Again, the angular velocity of joint 3 is equivalent to the rate of change of angular rotation of joint 1. There is only one rotation of the joint involved for the front end during the digging cycle. Thus, the angular velocity of joint 4 is also the same as the angular velocity of joint 1. Alternately, it can be stated that the whole front-end assembly gets the same rotation as the joint 1 during the digging cycle, and the angular velocity only involves the axis shift. Eqs. (8) through (18) define the forward kinematics of the shovel front end. The angular and linear velocities of the shovel front-end components are defined using these equations with known initial rotation and crowd-arm extension:

0 0 0 0

> 0 0

0 0 0 0

Therefore, the linear velocity of joint 1 is zero:

114 Lagrangian Mechanics

w

 wq *d v <sup>v</sup> dt*

1 10 1 10 1 10 0 1 1 10 0 1 1

is only an axis shift involved (from Z-axis to Y-axis) during the propagation:

2 21 2

 wq

w

\

22 2

& &&

, ,and

qq

2 1 1 22 1

ê ú = += ê ú

 q

*R Z*

ˆ

& &

 w

*v*

*d dt* w 0 0 0

é ù

ê ú ë û

> 0 0 0

é ù

ê ú ë û

ê ú = = ê ú

Eq. (12) is obtained from Eq. (4) for joint 1 (*i* = 0), a revolute joint. It is evident from this equation that the angular velocity of the first link is only around Z-axis and is equivalent to the rate of change of angular rotation around joint 1. The linear velocity propagation to joint 1 can be computed using Eq. (5) as Eq. (13). The first link experiences only the rotational motion.

ˆ ˆ

*R ZR Z* 0

For the prismatic joint 2 (*i* = 1), Eq. (6) computes the angular velocity of the link 2 (the crowd arm). The propagation of angular velocity to joint 2, as given in Eq. (14), shows that the angular velocity of joint 2 is dependent upon the rate of change of angular rotation of joint 1, and there

0

é ù

0

ê ú ë û

 q

*are zero*

ê ú = + == + = ê ú

& &

 wq

& (9)

& (11)

1

q

ê ú ë û

&

0

é ù

(10)

(12)

(13)

(14)

w

ê ú = = ê ú

0 0 0

é ù ê ú <sup>=</sup> ê ú ê ú ë û

$$\begin{aligned} \,^3o\_3 &= \begin{bmatrix} 0\\0\\\dot{\theta}\_1 \end{bmatrix} \end{aligned} \tag{16}$$

$$\mathbf{P}^3 \mathbf{v}\_3 = \,\_2^3 \mathbf{R} \left( \,^2 \mathbf{v}\_2 + \,^2 \mathbf{o}\_2 \,^2 \mathbf{X} \,^2 \mathbf{P}\_1 \right) = \begin{bmatrix} \mathbf{d}\_z \dot{\theta}\_1\\ \mathbf{a}\_1 \dot{\theta}\_1\\ \mathbf{0} \end{bmatrix} \tag{17}$$

$$\mathbf{A}^{4}\mathbf{v}\_{4} = \,^{4}\_{\flat}\mathbf{R}\left(\,^{3}\mathbf{v}\_{\flat} + ^{3}\mathbf{o}\_{\flat}\,^{3}\mathbf{X}^{\flat}\,^{\mathsf{P}}\mathbf{P}\_{4}\right) = \begin{bmatrix} \,^{4}\mathbf{d}\_{z}\dot{\theta}\_{1}\mathbf{c}\_{\flat} + \left(\mathbf{a}\_{\flat} + \mathbf{a}\_{z}\right)\dot{\theta}\_{1}\mathbf{s}\_{\flat} \\ \cdot\,^{4}\mathbf{d}\_{z}\dot{\theta}\_{\flat}\mathbf{s}\_{\flat} + \left(\mathbf{a}\_{\flat} + \mathbf{a}\_{z}\right)\dot{\theta}\_{\flat}\mathbf{c}\_{\flat} \\ \mathbf{0} \end{bmatrix} \tag{18}$$

$$T\_4^1 = \begin{bmatrix} c\_\beta & -s\_\beta & 0 & a\_1 + a\_2 c\_\beta \\ s\_\beta & c\_\beta & 0 & a\_2 s\_\beta - d\_2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{19}$$

*Inverse kinematics of cable shovel front*-*end assembly*: The inverse shovel kinematics determine the set of joint angles and the length for the crowd arm when the desired position and orientation of the shovel dipper are known in the reference coordinate frame 0. This inverse kinematic is useful when the dipper traverses a known trajectory to determine the joint rotation and crowdarm extension required to achieve this trajectory. An approach, similar to the one used by Wu [15] for the reverse kinematic model of cable shovel, is used to determine the crowd-arm extension and rotation with known trajectory points. The inverse kinematic model can be achieved by coordinate transformations to obtain the dipper coordinate in coordinate frame 4, relative to coordinate frame 1. The modifications of the transformation matrix equations result in Eqs. (19), (20), and (21):

$$T\_4^1 = \left[T\_1^0\right]^{-1} T\_4^0 \tag{20}$$

$$T\_4^0 = \begin{bmatrix} r\_{11} & r\_{12} & r\_{13} & p\_x \\ r\_{21} & r\_{22} & r\_{23} & p\_y \\ r\_{31} & r\_{32} & r\_{33} & p\_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{21}$$

The individual matrix elements are given as follows:

$$\begin{aligned} &r\_{11}: \mathbf{c}\_{1}\mathbf{c}\_{\beta} - s\_{1}\mathbf{s}\_{\beta}, \\ &r\_{12}: -\mathbf{c}\_{1}\mathbf{s}\_{\beta} - s\_{1}\mathbf{c}\_{\beta}, \\ &r\_{13}: \mathbf{0}; \\ &r\_{21}: \mathbf{s}\_{1}\mathbf{c}\_{\beta} + \mathbf{c}\_{1}\mathbf{s}\_{\beta}, \\ &r\_{22}: -\mathbf{s}\_{1}\mathbf{s}\_{\beta} + \mathbf{c}\_{1}\mathbf{c}\_{\beta}, \\ &r\_{23}: \mathbf{0}; \\ &r\_{31}: \mathbf{0}; \\ &r\_{33}: \mathbf{0}; \\ &P\_{x}: \mathbf{a}\_{2}\mathbf{c}\_{\mathbf{c}}\mathbf{c}\_{\beta} - \mathbf{a}\_{2}\mathbf{s}\_{\mathbf{c}}\mathbf{s}\_{\beta} + \mathbf{a}\_{1}\mathbf{c}\_{1} + \mathbf{d}\_{2}\mathbf{s}\_{1}; \\ &P\_{y}: \mathbf{a}\_{2}\mathbf{s}\_{\mathbf{c}}\mathbf{c}\_{\beta} + \mathbf{a}\_{2}\mathbf{c}\_{\mathbf{c}}\mathbf{s}\_{\beta} + \mathbf{a}\_{1}\mathbf{s}\_{1} - \mathbf{d}\_{2}\mathbf{c}\_{1}; \\ &P\_{z}: \mathbf{0} \end{aligned}$$

Here, (*p*x, *p*y, and *p*z) are the coordinates of the dipper tip in the reference coordinate frame 0. Eq. (22) can be derived from Eq. (2) and Eq. (23) from Eq. (20). Comparing the individual matrix elements on both sides of Eq. (23) and using simple arithmetic and trigonometric operations, the crowd-arm extension and rotation can be computed using Eqs. (25) and (26), respectively. The inverse kinematic model can be used to compute the positions and velocities of individual links and joints of the front-end assembly for a known trajectory:

$$
\begin{bmatrix} \begin{bmatrix} \mathbf{T}\_1^0 \end{bmatrix} \end{bmatrix}^{-1} = \begin{bmatrix} \mathbf{c}\_1 & \mathbf{s}\_1 & \mathbf{0} & \mathbf{0} \\ -\mathbf{s}\_1 & \mathbf{c}\_1 & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & 1 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & 1 \end{bmatrix} \tag{22}
$$

#### Mechanics of Electric Rope Shovel Performance and Reliability in Formation Excavation http://dx.doi.org/10.5772/65333 117

$$
\begin{bmatrix}
\end{bmatrix} = \begin{bmatrix}
\mathbf{c}\_\beta & -\mathbf{s}\_\beta & \mathbf{0} & \mathbf{a}\_1 + \mathbf{a}\_2 \mathbf{c}\_\beta \\
\mathbf{s}\_\beta & \mathbf{c}\_\beta & \mathbf{0} & \mathbf{a}\_2 \mathbf{s}\_\beta - \mathbf{d}\_2 \\
\mathbf{0} & \mathbf{0} & 1 & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & 1
\end{bmatrix} \tag{23}
$$

$$d\_2 = a\_2 \mathbf{s}\_\beta + \sqrt{p\_x^2 + p\_y^2 - a\_1^2 - a\_2^2 - 2a\_1 a\_2 \mathbf{c}\_\beta + \left(a\_2 \mathbf{s}\_\beta\right)^2} \tag{24}$$

$$\theta\_1 = A \tan 2 \left( a\_2 s\_\rho - d\_z \right.\\ \left. \pm \sqrt{p\_x^2 + p\_y^2 - \left( a\_2 s\_\rho - d\_z \right)^2} \right) - A \tan 2(p\_y, p\_x) \tag{25}$$
