aspects stem from evaluating posture stress in real time (ergonomic aspects), the cognitive state [Camblor et al., 2022] and the expertise of the operators [Benhabib et al., 2020].

In the same way, the MPC can straightforwardly be formulated to follow a previ-
ously generated trajectory instead of a target pose. This would allow for path plan-
ning algorithms to be plugged right before the controller, integrating unexplored
behaviors like, for example, human-aware collision avoidance.

## 6.2 Closing Remarks ↑

Collaborative robots constitute a fundamental piece in the ongoing modern indus-
trial transformation towards human-centric and sustainable manufacturing. As
such, the market demand for cobots has seen an increasing growth trend recently in
market cap and is projected to even accelerate development for the future [Gambao,
2023]. Even though the cobot industry represents a small part of the overall robotics
market cap, it still constitutes a young market with immense potential in the world
and European market. Notably, Europe is the second main consumer of cobots in
the world (after Asia) and is home to many of the biggest manufacturers.

Remarkably, the versatility and rapid adoption of cobots (as opposed to industrial robots) shows the potential to accelerate the development of multiple industrial domains. In this context, the continued investment in innovative ways to integrate cobots into every industry should be considered a strategic priority for industrial development.

Since their inception, cobots marked a hiatus from previous robot design trends to
prioritize the new and (thus far) unconceived scenario of collaboration between hu-
mans and robots. These promises, while ambitious and enticing, have been ahead of
the control strategies required to actually enable true collaboration.

The present work constitutes an effort towards showcasing and extending cobots flexibility in collaborative and dynamic environments. It shows that the promises of true human-robot collaboration demand a significant design effort. From the hardware point of view, robots are required to evolve (as introduced with cobots). From a control perspective, human-centric strategies demand an equally significant departure from classical control approaches. In the end, these are fundamental steps towards the true collaboration that will enable augmenting the human-robot synergy.
# Appendix A

## Quadratic Programming

Quadratic Programming is a technique that permits optimizing some vector variable $x$ with respect to some second order cost function $f(x)$ and some linear equality and inequality constraints (lower/upper bounds).

The canonical QP optimization problem is formulated as:

$$ 
\begin{array}{ll}
x^{\text{opt}} = \arg \min_{x} & \frac{1}{2} x^T Q x + g^T x + r \\
\text{s.t.} & C_l \leq C_i x \leq C_u \\
& b_l \leq x \leq b_u \\
& C_c x = d
\end{array}
\qquad (A.1)
$$

where:

* $x \in \mathbb{R}^{n \times 1}$ designates the optimizing variable (also referred to as the decision variable).

* $Q \in \mathbb{R}^{n \times n}$ contains the quadratic cost function coefficients.

* $g \in \mathbb{R}^{1 \times n}$ refers to the linear coefficients, also referred to as the gradient vector.

* $r \in \mathbb{R}$ refers to a constant cost term, independent of the decision variable, and so it can be neglected, as it doesn't affect the result of the optimization.

* The inequality constraints are expressed as a linear system described with the lower and upper bounds vectors $C_l, C_u$ and the coefficients matrix $A$ that relates them to the optimizing variable. This constitutes the most general set of constraints and is often used to formulate the other ones. Some solvers only offer this alternative (and equivalent) formulation:

$$ \begin{bmatrix} C_i \\ -C_i \end{bmatrix} x \le \begin{bmatrix} C_u \\ -C_l \end{bmatrix} \qquad (A.2) $$

* The equality constraints are expressed as a linear system described with the vector $d$ and the coefficients matrix $C$ that relates them to the optimizing variable. They can be embedded into the inequality constraints by employing $d$ both as the lower and upper bounds.
* Finally, the variable bounds express the decision variable lower $b_l$ and upper $b_u$ bounds. These can be included into the inequality constraints by adding an identity matrix to $C$.

## Embedded Constraints

Often, Quadratic Program (QP) solvers offer only the linear inequality constraints, as the others can be embedded in them for simplification:

$$ \begin{bmatrix} b_l \\ A_l \\ d \end{bmatrix} \le \begin{bmatrix} I_n \\ A \\ C_e \end{bmatrix} x \le \begin{bmatrix} b_u \\ A_u \\ d \end{bmatrix} \qquad (A.3) $$

## Simplified formulation

Then, it is possible to arrive at a simplified canonical formulation:

$$ \begin{array}{ll} x^{opt} = \arg\min_{x} & \frac{1}{2}x^TQx + g^T x \\ \text{s.t.} & A_l \le Ax \le A_u \end{array} \qquad (A.4) $$

## A.1 Weighted Least Square ↑

The following least-square optimization problem can be reformulated in a standard quadratic problem form as in (A.1):

$$ \arg\min_{x} \quad ||Ex - b||_{W}^{2} \qquad (A.5) $$

where $W$ is a positive semi-definite matrix containing weights the optimization variable.

Given:

$$ \|Ex - b\|_{W}^{2} = (Ex - b)^{T}W(Ex - b) \qquad (A.6) $$

$$ = x^T E^T W E x - 2b^T W E x + b^T b \qquad (A.7) $$

then we can express (A.5) in the initial formulation of the QP optimization (A.1):

$$ Q = 2E^TWE, \quad g^T = -2b^TWE, \quad r = b^Tb \qquad (A.8) $$

A commonly used case of this least squares optimization is for when the weight matrix is an identity $W = I$ (giving equal weight to all elements in the decision variable):

$$ \arg\min_{x} \quad ||Ex - b||^{2} \qquad (A.9) $$
# Appendix B

## Discretization of Continuous LTI systems

A dynamic system can be parameterized with state $x \in \mathbb{R}^n$ and input $u \in \mathbb{R}^m$ vectors (where n is the size of the state and m the size of the input variables). The state vector is the smallest subset of system variables that contains all the information required to describe the system at some time instant. The input variable corresponds to the decision variable used to drive the system to some desired or target state.

Given some non-linear system dynamics described with $f(x, u)$, the continuous state-space system representations is:

$$ \dot{x}(t) = A_c(t)x(t) + B_c(t)u(t) \quad (B.1) $$

$$ y(t) = C(t)x(t) + D(t)u(t) \quad (B.2) $$

with:

$$ A_c(t) = \frac{\delta f(x, u)}{\delta x} \qquad B_c(t) = \frac{\delta f(x, u)}{\delta u} \quad (B.3) $$

* $\dot{x}$: represent the system state and its derivative

* $u$: represents the system input

* $y$: represents the system observed state

where:

* $A_c, B_c$: LTI matrices that express the relation between the system state derivative and its state and input

* $C, D$: LTI matrices that express the relation between the system observed state and its state and input

The solution to the differential equation is:

$$ x(t) = e^{A_c t} x(0) + \int_0^t e^{A_c(t-\tau)} B_c(\tau) \delta\tau \quad (B.4) $$
## B.1 Discrete-time State-space ↑

The time-discrete equivalent of (B.1) and (B.2) are:

$$x_{k+1} = A_d x_k + B_d u_k \quad (B.5)$$

$$y_k = C x_k + D u_k \quad (B.6)$$

Note that for the general case of a non-linear system, all these matrices also depend on the state. This means that, the system needs to be linearized at some time instant $t_n$ and then $A_{d,n}$ can be assumed constant for a limited time window. The same conclusions can be applied for $B, C, D$.

Using (B.4) with a sampling period of $T_s$, we arrive at the exact solutions:

$$A_d = e^{A_c T_s} \quad (B.7)$$

$$B_d = \int_0^{T_s} e^{A_c \tau} \delta \tau B_c = K B_c \quad (B.8)$$

$$\text{with } K = A_c^{-1} (e^{A_c T_s} - I_n) \quad (B.9)$$

It is important to note that (B.8) is defined even if $A_c^{-1}$ does not ($A_c$ not reversible), as shown by developing the expressions through their taylor series:

$$A_d = I_n + A_c T_s + \frac{(A_c T_s)^2}{2} \dots = \sum_{n=0}^{\infty} \frac{(A_c T_s)^n}{n!} \quad (B.10)$$

$$K = A_c^{-1} \left( \sum_{n=0}^{\infty} \frac{(A_c T_s)^n}{n!} - I_n \right) \quad (B.11)$$

$$= A_c^{-1} (I_n + A_c T_s + \frac{(A_c T_s)^2}{2} \dots - I_n) \quad (B.12)$$

$$= A_c^{-1} \sum_{n=1}^{\infty} \frac{(A_c T_s)^n}{n!} \quad (B.13)$$

$$= \sum_{n=1}^{\infty} \frac{A_c^{n-1} T_s^n}{n!} \quad (B.14)$$
# Appendix C

## Summary: Velocity-based Pose Tracking as a Quadratic Program

The inverse kinematic problem consists of finding the robot configuration satisfying some operational-space position and orientation. This section introduces a way to solve this problem for a robot that admits being controlled by its joint velocities.

This controller is similar to the one in [Joseph et al., 2020]. It is formulated as a linearly constrained Quadratic Program (QP) that optimizes for the joint velocity; the main difference being the added generic linear constraints in (C.5) that will be detailed shortly. The resulting QP formulation is:

$$ \dot{\mathbf{q}}^{\text{opt}} = \arg \min_{\dot{\mathbf{q}}} f_{\text{main}}(\dot{\mathbf{q}}) + w_{\text{reg}} f_{\text{reg}}(\dot{\mathbf{q}}) \quad (C.1) $$

s.t.

$$ \dot{\mathbf{q}}_m \le \dot{\mathbf{q}} \le \dot{\mathbf{q}}_M \quad (C.2) $$

$$ \dot{\mathbf{q}} \in C_q \quad (C.3) $$

$$ \dot{\mathbf{q}} \in C_{\ddot{\mathbf{q}}} \quad (C.4) $$

$$ A_l \le A \dot{\mathbf{q}} \le A_u \quad (C.5) $$

$$ \dot{\mathbf{q}}, \dot{\mathbf{q}}_m, \dot{\mathbf{q}}_M \in \mathbb{R}^n $$

with:

$$ C_q = \left\{ \dot{\mathbf{q}} \in \mathbb{R}^n \mid \dot{\mathbf{q}} \in \left[ \frac{\mathbf{q}_m - \mathbf{q}_k}{\Delta t}, \frac{\mathbf{q}_M - \mathbf{q}_k}{\Delta t} \right] \right\} \quad (C.6) $$

$$ C_{\ddot{\mathbf{q}}} = \{ \dot{\mathbf{q}} \in \mathbb{R}^n \mid \dot{\mathbf{q}} \in [\dot{\mathbf{q}}_k + \ddot{\mathbf{q}}_m \Delta t, \dot{\mathbf{q}}_k + \ddot{\mathbf{q}}_M \Delta t] \} \quad (C.7) $$

$$ f_{\text{main}}(\dot{\mathbf{q}}) = \| {}^B\mathbf{v}^* - {}^B\mathbf{J}(\mathbf{q}_k)\dot{\mathbf{q}}\|_2^2 \quad (C.8) $$

$$ {}^B\mathbf{v}^* = K_p {}^B\mathbf{e} + {}^B\mathbf{v}^r \quad \mathbf{e}, \mathbf{v} \in \mathfrak{SE}(3) \quad (C.9) $$

$$ {}^B\mathbf{e} = \log(\mathcal{X}_k^{-1} \mathcal{X}^r) \quad \mathcal{X} \in \mathbb{SE}(3) \quad (C.10) $$

$$ f_{\text{reg}}(\dot{\mathbf{q}}) = \|\dot{\mathbf{q}}_{\text{reg}} - \dot{\mathbf{q}}\|_2^2 $$

$$ \dot{\mathbf{q}}_{\text{reg}} = K_q (\mathbf{q}_{\text{reg}} - \mathbf{q}) \quad (C.10) $$

$$ \mathbf{q}_{\text{reg}} = \frac{(\mathbf{q}_M - \mathbf{q}_m)}{2} $$
where B implies the corresponding quantity is expressed in the body frame (see Section 3.2.4). Additionally:

* $\dot{q}^{opt}$ corresponds to the optimal $\dot{q}$ that minimizes the cost function; this implies it will be applied in the next step as $\dot{q}_{k+1}$. On the other hand, the current states of the robot referred to as $\mathcal{X}_k, q_k, \dot{q}_k$ and are used in (C.6), (C.7), (C.8), (C.9) and (C.9).

* $f_{\text{main}}(\dot{q})$ in (C.8) designates the main task cost function. In this case, it corresponds to a pose tracking task that minimizes the distance between some desired twist $v^*$ (see (C.9) explained below) and the robot current twist, which is computed as a function of the optimizing variable (the joint velocities) through the robot Jacobian $J(q)$.

* $v^*$ from (C.8) is defined in (C.9) as a Proportional Derivative (PD) from a trajectory providing a reference pose $\mathcal{X}^r$ and twist $v^r$. The proportional gain $K_p$ provides a way to modulate the correction from the error term. Finally, $e$ is constructed as the desired twist displacement required to correct the robot pose from $\mathcal{X}$ to $\mathcal{X}^r$.

* $f_{\text{reg}}(\dot{q})$ is a regularisation task or function that helps uniquely determine an optimal solution in redundant robots by acting on the extra degrees of freedom. In this case, the task tries to maintain a configuration close to the robot mean position relative to its bounds. The details can be found in [Joseph et al., 2020]. Then $w_{\text{reg}}$ is chosen small enough not to affect the main task.

* $C_q$ in (C.6) offers one way to implement (C.3) via the first order taylor expansion on the current robot configuration $q_k$, establishing a relation between the joint bound limits $(q_m, q_M)$ and the joint velocities.

* Similarly, $C_{\ddot{q}}$ in (C.7) is one possible implementation of (C.4); in this case, it exploits the first order taylor expansion of the current joint velocity $\dot{q}_k$ to incorporate the joint acceleration limits $(\ddot{q}_m, \ddot{q}_M)$ as a function of the decision variable, the joint velocities.

* Finally, (C.5) offers a way to incorporate linear constraints. For instance, it can be used to express workspace-related limits like the geometric space or, for example, the twist constraints can be formulated as:

$$v_{b1} \le J\dot{q} \le v_{bu} \qquad (C.11)$$
# Appendix D

## Summary: Torque-based Pose Tracking as a Quadratic Program

An inverse kinematics solver as a *Quadratic Program (QP)* is presented in *Appendix C*. This section extends the previous strategy to formulate an inverse dynamics solver. While both problems involve working backwards to solve for unknowns, inverse kinematics focuses on the geometric view of the problem, solving for the joint angles necessary to achieve a certain end-effector pose, ignoring the required forces or torques. In contrast, inverse dynamics deals with the problem of finding the torques and forces required for an end-effector motion. This accounts for the system dynamic properties such as the mass, inertia, external forces, etc.

The controller presented here is similar to the one in [Joseph et al., 2018b]. It is designed for torque-controlled robots. By exploiting the robot dynamics model (see Section 4.2.1), we can formulate the joint accelerations as a function of the joint torques. Furthermore, the Cartesian spatial acceleration can then be related to the
joint accelerations through the robot Jacobian, enabling a QP formulation of the task-space tracking problem:

$$ \tau^{\text{opt}} = \arg\min_{\tau} f_{\text{main}}(\tau) + w_{\text{reg}} f_{\text{reg}}(\tau) \qquad (\text{D.1}) $$

s.t.

$$ \tau_m \le \tau \le \tau_M \qquad (\text{D.2}) $$

$$ \tau \in C_q \qquad (\text{D.3}) $$

$$ \tau \in C_{\dot{q}} \qquad (\text{D.4}) $$

$$ \tau \in C_{\dot{\tau}} \qquad (\text{D.5}) $$

$$ A_l \le A\dot{q} \le A_u \qquad (\text{D.6}) $$

$$ \tau, \tau_m, \tau_M \in \mathbb{R}^n $$

with:

$$ \ddot{q}(\tau) = M(q_k)^{-1}(\tau + b(q_k, \dot{q}_k)) \qquad (\text{D.7}) $$

$$ C_q = \left\{ \ddot{q}(\tau) \in \mathbb{R}^n \mid \ddot{q}(\tau) \in \left[ \frac{2(q_m - q_k - \dot{q}_k \Delta t)}{\Delta t^2}, 2\frac{(q_M - q_k - \dot{q}_k \Delta t)}{\Delta t^2} \right] \right\} \qquad (\text{D.8}) $$

$$ C_{\dot{q}} = \left\{ \ddot{q}(\tau) \in \mathbb{R}^n \mid \ddot{q}(\tau) \in \left[ \frac{(\dot{q}_m - \dot{q}_k)}{\Delta t}, \frac{(\dot{q}_M - \dot{q}_k)}{\Delta t} \right] \right\} \qquad (\text{D.9}) $$

$$ C_{\dot{\tau}} = \{ \tau \in \mathbb{R}^n \mid \tau \in [\tau_k + \dot{\tau}_m \Delta t, \tau_k + \dot{\tau}_M \Delta t] \} \qquad (\text{D.10}) $$

$$ f_{\text{main}}(\tau) = \| {}^B\dot{v}^* - \dot{v}(\tau) \|_2^2 \qquad (\text{D.11}) $$

$$ \dot{v}(\tau) = J(q_k) \ddot{q}(\tau) \qquad (\text{D.11}) $$

$$ {}^B v^* = K_p {}^B e + K_d {}^B \dot{e} + {}^B \dot{v}^r \qquad e, \dot{e}, v, \dot{v} \in \mathfrak{SE}(3) \qquad (\text{D.12}) $$

$$ {}^B e = \log(\chi_k^{-1} \chi^r) \qquad \chi \in \mathcal{S} \mathcal{E}(3) \qquad (\text{D.12}) $$

$$ {}^B \dot{e} = {}^B v^r - {}^B v_k \qquad (\text{D.12}) $$

$$ f_{\text{reg}}(\tau) = \| \tau_{\text{reg}} - \tau \|_2^2 \qquad (\text{D.13}) $$

where B implies the corresponding quantity is expressed in the *body frame* (see Section 3.2.4). Additionally:

* $\tau^{\text{opt}}$ corresponds to the optimal $\tau$ that minimizes the cost function; this implies it will be applied in the next step as $\tau_{k+1}$. On the other hand, the current states of the robot referred to as $\chi_k, q_k, \dot{q}_k, \tau_k$ and are used in (D.7), (D.8), (D.10), (D.11) and (D.12).

* $f_{\text{main}}(\tau)$ in (D.11) designates the main task cost function. In this case, it corresponds to a pose tracking task that minimizes the distance between some desired twist derivative $\dot{v}^*$ (analogous to a rigid body motion acceleration, see (D.12) explained below) and the robot current spatial acceleration, which is computed as a function of the optimizing variable (the joint torques) through the robot dynamics model in (D.7).

* $v^*$ from (D.11) is defined in (D.12) as a Proportional Derivative (PD) term from
a trajectory providing a reference pose $\mathcal{X}^r$, a twist $v^r$ and a spatial acceleration $\dot{v}^r$ (the latter used as a feed forward term). The proportional gain $K_p$ provides a way to modulate the correction from the error term $e$, which is constructed as the desired twist displacement required to correct the robot pose from $\mathcal{X}$ to $\mathcal{X}^r$; in the same way, the derivative gain $K_d$ modulates the twist error $\dot{e}$.

* $f_{\text{reg}}(\tau)$ is a regularisation task or function that helps uniquely determine an optimal solution in redundant robots by acting on the extra degrees of freedom. In this case, the task tries to maintain a configuration close to the robot mean position relative to its bounds without affecting the main task (because $w_{\text{reg}}$ is chosen small enough not to affect it). Then $\tau_{\text{reg}}$ can be chosen in different ways depending on the targeted application; here, it is chosen to ensure the convergence of the robot configuration to a neutral configuration $q_{\text{reg}}$. The servoing of this configuration can be expressed at the joint torque level using a PD controller such that:

$$ \tau_{\text{reg}} = K_q(q_{\text{reg}} - q) - K_{\dot{q}}\dot{q} \quad (\text{D.14}) $$

with $K_q$ and $K_{\dot{q}}$ positive proportional and derivative gains.

* $C_q$ in (D.8) offers one way to implement (D.3) via the second order taylor expansion on the current robot configuration $q_k$, establishing a relation between the joint bound limits ($q_m, q_M$) and the joint accelerations (and indirectly, the joint torques).

* Similarly, $C_{\dot{\tau}}$ in (D.10) is one possible implementation of (D.5); in this case, it exploits the first order taylor expansion of the current joint torque $\tau_k$ to incorporate the joint torque derivative limits ($\dot{\tau}_m, \dot{\tau}_M$) as a function of the decision variable, the joint torque.

* Finally, (D.6) offers a way to incorporate linear constraints. For instance, it can be used to express workspace-related limits like the geometric space or, for example, the spatial acceleration constraints can be formulated as:

$$ \dot{v}_{b_l} \le J\ddot{q}(\tau) \le \dot{v}_{b_u} \quad (\text{D.15}) $$

Given the interpolated first step of the trajectory computed by the Model Predictive Control (MPC), task space inverse dynamics has to be performed to compute the input joint torque for the robot.

Feeding the inverse dynamics solver with the desired acceleration computed by the MPC may sound tempting. Nevertheless, the feedback provided by the closed-loop MPC is, partly due to the interpolated output, not efficient enough to properly reject tracking errors related to the inaccuracies in the model of the robot.

These inaccuracies are mostly related to dry friction at the joint level, imperfectly rejected by the lower-level torque control loop in most robots. As a consequence, a PD controller including a feed-forward term in acceleration is used to compute a corrected desired acceleration.
For convenience and to further clarify how the different components are connected, this Section revisits some of the concepts previously introduced in Section 4.2.2:

$$ \dot{v}^* = K_p \log(\mathcal{X}^{-1}\mathcal{X}_{\text{mpc}}) + K_d(\nu_{\text{mpc}} - \nu) + \dot{v}_{\text{mpc}} \quad (\text{D.16}) $$

where $\mathcal{X}_{\text{mpc}}, \nu_{\text{mpc}}, \dot{v}_{\text{mpc}}$ are respectively the interpolated desired pose, twist and acceleration outputted by the MPC; $\mathcal{X}, \nu$ are the measured pose and twist of the robot and $K_p$ and $K_d$ are positive proportional and derivative gains.

Given this control acceleration, one can solve task-space inverse dynamics under constraints through a QP formulation, similar to the one in [Joseph et al., 2018b].
# Appendix E

## Smooth stopping trajectory

The smooth stopping analysis contemplates two stages in the trajectory, as depicted in Figure 5.19 and Table 5.6.

### E.1 Acceleration Reversing Stage ↑

When the initial acceleration and velocity are in the same sens. This considers the two following starting conditions:

$$ \textcircled{1} = \left\{ \begin{array}{l} 0 \le a_k \le a_M, \\ 0 \le v_k \le v_M, \end{array} \right. \qquad \textcircled{2} = \left\{ \begin{array}{l} a_M \le a_k \le 0, \\ v_M \le v_k \le 0 \end{array} \right. \qquad (\text{E.1}) $$

yielding:

$$ j_R(t) = \begin{cases} 0 & t=0 \\ j_M & 0 < t < t_R \\ 0 & t = t_R \end{cases} \qquad (\text{E.2}) $$

$$ a_R(t) = \begin{cases} -a_k & t=0 \\ -a_k + j_M t & 0 < t < t_R \\ 0 & t = t_R \end{cases} \qquad (\text{E.3}) $$

$$ v_R(t) = \begin{cases} -v_k & t=0 \\ -v_k - a_k t + \frac{j_M t^2}{2} & 0 < t \le t_R \end{cases} \qquad (\text{E.4}) $$

$$ \Delta p_R(t) = \begin{cases} 0 & t=0 \\ -v_k t - \frac{a_k t^2}{2} + \frac{j_M t^3}{6} & 0 < t \le t_R \end{cases} \qquad (\text{E.5}) $$

and finally:

$$ t_R = \frac{a_k}{j_M}, \quad v_R(t_R) = -v_k - \frac{a_k^2}{2j_M} \qquad (\text{E.6}) $$

$$ \rightarrow \Delta v_R(t_R) = v_k - v_R(t_R) \qquad (\text{E.7}) $$

$$ \rightarrow \Delta p_R(t_R) = -v_k \frac{a_k}{j_M} - \frac{2a_k^3}{6j_M^2} \qquad (\text{E.8}) $$
## E.2 Deceleration Stage ↑

When the initial acceleration is zero or in the opposite sense of the initial velocity. Note that if an acceleration reversing stage was necessary, then $a_k = 0, v_k = v_R(t_R)$.

$$j_D(t) = \begin{cases} 0 & t=0 \\ j_M & 0 < t < t_1 \\ 0 & t_1 \le t < t_{12} \\ -j_M & t_{12} \le t < t_{123} \\ 0 & t_{123} \le t \end{cases} \qquad (E.9)$$

$$a_D(t) = \begin{cases} a_k & t=0 \\ a_k + j_M t & 0 < t < t_1 \\ a_M & t_1 \le t \le t_{12} \\ a_M - j_M t & t_{12} < t < t_{123} \\ 0 & t_3 \le t \end{cases} \qquad (E.10)$$

$$v_D(t) = \begin{cases} -v_k & t=0 \\ -v_k + a_k t + \frac{i_M t^2}{2} & 0 < t \le t_1 \\ v_D(t_1) + a_M t & t_1 < t \le t_{12} \\ v_D(t_{12}) + a_M t - \frac{i_M t^2}{2} & t_{12} < t < t_{123} \\ 0 & t_{123} \le t \end{cases} \qquad (E.11)$$

$$\Delta p_D(t) = \begin{cases} 0 & t=0 \\ -v_k t - \frac{i_M t^3}{6} & 0 < t \le t_1 \\ \Delta p_D(t_1) + v_D(t_1) t + \frac{a_M t^2}{2} & t_1 < t \le t_{12} \\ \Delta p_D(t_{12}) + v_D(t_{12}) t + \frac{a_M t^2}{2} - \frac{i_M t^3}{6} & t_{12} < t \le t_{123} \\ \Delta p_D(t_{123}) & t_{123} < t \end{cases} \qquad (E.12)$$
where:

$$t_{12} = t_1 + t_2, \quad t_{123} = t_1 + t_2 + t_3, \quad t_1 = \frac{a_M - a_k}{j_M}, \quad t_3 = \frac{a_M}{j_M} \qquad (E.13)$$

$$\Rightarrow \Delta v_D(t_{123}) = \int_0^{t_{123}} a_D(t) dt - v_k \qquad (E.14)$$

$$= a_k t_1 + \frac{j_M t_1^2}{2} + a_M t_{12} + a_M t_{123} - \frac{j_M t_{123}^2}{2} \qquad (E.15)$$

$$\Rightarrow \Delta p_D(t_{123}) = \Delta p_D(t_{12}) + v_D(t_{12}) t_{123} + \frac{a_M t_{123}^2}{2} - \frac{j_M t_{123}^3}{6} \qquad (E.16)$$

$$= \underbrace{\Delta p_D(t_1) + v_D(t_1) t_{12} + \frac{a_M t_{12}^2}{2}}_{\Delta p_D(t_{12})} + \underbrace{(v_D(t_1) + a_M t_{12}) t_{123}}_{v_D(t_{12})} + \frac{a_M t_{123}^2}{2} - \frac{j_M t_{123}^3}{6} \qquad (E.17)$$

$$= \underbrace{-v_k t_1 - \frac{j_M t_1^3}{6}}_{\Delta p_D(t_1)} + \underbrace{(v_D(t_1) + a_M t_1) t_{12}}_{v_D(t_1)} + \frac{a_M t_{12}^2}{2} + \frac{a_M t_{123}^2}{2} - \frac{j_M t_{123}^3}{6} + \underbrace{(-v_k + a_k t_1 + \frac{j_M t_1^2}{2} + a_M t_{12}) t_{123}}_{v_D(t_1)} \qquad (E.18)$$

$$= -v_k t_1 - \frac{j_M t_1^3}{6} + \underbrace{(-v_k t_1 - \frac{j_M t_1^3}{6} + a_M t_1) t_{12}}_{v_D(t_1)} + \frac{a_M t_{12}^2}{2} + \frac{a_M t_{123}^2}{2} - \frac{j_M t_{123}^3}{6} + \underbrace{(-v_k + a_k t_1 + \frac{j_M t_1^2}{2} + a_M t_{12}) t_{123}}_{v_D(t_1)} \qquad (E.19)$$

Two cases considered:

$$\textcircled{1} \quad 0 \le a_k \le a_m, \quad v_k \le \frac{2a_m^2 - a_k^2}{2j_M} \qquad (E.20)$$

$$t_3 = \frac{a_m}{j_m}, \quad t_1 = \frac{a_m - a_k}{j_m} \qquad (E.21)$$

Only a triangular trajectory is required.

$$\textcircled{2} \quad 0 \le a_k, \quad v_k > \frac{2a_m^2 - a_k^2}{2j_M} \qquad (E.22)$$

A full TAP trajectory is required.

(E.23)
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