**4. Dynamic model of the cable shovel front-end assembly**

4, relative to coordinate frame 1. The modifications of the transformation matrix equations

ë û (20)

(21)

(22)

<sup>1</sup> 1 00 *T TT* <sup>414</sup> - = é ù

11 12 13 0 21 22 23

*rrr p <sup>T</sup> rrr p*

11 1 1 12 1 1

*r cc ss r cs sc*

b

: ; : ;


b

 b

> b

> > b

 b

21 1 1 22 1 1

b

*r sc cs r ss cc*

: ; : ;

+ - +

b

b

b

: 0;

: 0; : 0; : 0; : 0;

: 0

21 21 11 21 21 21 11 21


: ; : ;

 b

> b

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

1 1

c s 00 s c 00

é ù ê ú

> 0 010 0 0 01

ê ú ë û

<sup>1</sup> <sup>0</sup> 1 1

 ê ú é ù <sup>=</sup> ë û ê ú

*P a cc a ss ac d s P a sc acs as d c*

13

*r*

*r r r r*

> *x y z*

*P*

links and joints of the front-end assembly for a known trajectory:

1


T

*rrr p*

é ù ê ú = ê ú ê ú ê ú ë û

*x y z*

31 32 33 0001

4

The individual matrix elements are given as follows:

result in Eqs. (19), (20), and (21):

116 Lagrangian Mechanics

The dynamic model defines forces and torques acting on the shovel links and joints from the kinematics parameters, such as accelerations. The forces require the computation of angular and linear accelerations, which can be obtained by time integration of the angular and linear velocities computed in the kinematic model. In its general form, the dynamic model can be defined as in Eq. (26) from Frimpong et al. [16]:

$$\begin{aligned} \text{D(\Theta)\ddot{\Theta}} + \text{C}\{\Theta, \dot{\Theta}\}\dot{\Theta} + \text{G}(\Theta) &= \text{F} - \text{F}\_{\text{load}}(\text{F}\_{\text{t}}, \text{F}\_{\text{n}})\\ \text{D(\Theta)} &= \text{mass matrix} \\ \text{C}\{\Theta, \dot{\Theta}\} &= \text{centrifugal and Corrilolis terms} \\ \text{G}(\Theta) &= \text{gravity terms} \end{aligned} \tag{26}$$

This dynamic model for a shovel is built using the Newton-Euler method and the position, velocity, and acceleration relationships computed from the kinematic model. The Newton-Euler dynamic algorithm for computing the crowd force and the hoist torque comprises of the following steps:

1. Compute the angular acceleration of every link in the forward direction, starting from the saddle and moving outward toward the last link (the dipper).

2. Compute the acceleration of every link in the system in the forward direction.

3. Compute the acceleration at the left of mass (centroid) of every link in the system in the forward direction.

4. Determine the force (F*<sup>i</sup>* ) acting on every link at the centroid of the link using and mass of the link.

5. Compute the joint torque (N*<sup>i</sup>* ) for every link.

The force and torque are computed at the centroid of each link. Therefore, the velocity and the acceleration of the centroid are computed for every link.
