## 6.6. Taylored Controllers for Subclasses of (1)

The control problem has often been tackled by focusing on particular subclasses of robot-object systems, which usually simplify the set of analysed trajectories, hence simplify (26) and (27). Let us provide some of them:

*   **Hopping robots:** most of the studies are based on compliant contact [349, 350, 189, 351], hence can be recast into a complementarity framework, see section 3.6.1. In view of the material in the foregoing sections, would it be possible to use of the stability results in [151] for a bouncing ball with moving base, and of the equivalence between the one degree-of-freedom juggler and the vertical hopper, to design a controller for the hopper based on internal sinusoidal excitation ? The preparation for the airborne phase for a 3-link planar biped is studied in [352, 353]. Models with an actuated foot are analyzed in [354]. Hopping robots may be inertially driven [189]. Mirror controllers [328, 278, 59, 164] apply to jugglers, thus also to hoppers as long as their dynamics are identical (see section 3.3.3).

*   **Running robots:** this is quite similar to hopping, excepted that the system has a forward motion with "steps" and a forward velocity. Airborne-phase control can be used to prepare for the next mode (configuration and velocity pre-impact shaping) [126]. The trajectory's apex during airborne phases is an important control objective. Also as mentioned above the cat-like strategy can be used for systems with conserved quantities.

*   **The mirror law for jugglers:** this has been proposed in [328], applied in [278, 59]. The principle is to track the object's motion and to design a mirror-like motion of the object's motion for the robot, and regulating the object at impacts. Similarly to the OGY method, this seems to mainly apply to juggling systems which satisfy $M_2(q) \gg M_1(q)$ (hence $\dot{q}_2$ remains continuous at impacts, an assumption that is widely spread in juggling systems analysis [355, 164, 278, 59, 163, 223, 141, 356, 70, 296, 348, 140]). The dead-beat controller designed in [13, 14] is shown to reduce to a mirror algorithm when this assumption holds.

*   **Systems with joint clearance (dynamical impact model):** the goal may be to compensate for the mechanical play using passive mechanisms (guaranteeing contact inside the joints), or using active control to guarantee persistent contact (backlash-free control) [3, 357, 358], including control constraints (5). Control redundancy plays a significant role in these approaches. The analysis of robustness properties of no-clearance controllers is studied for PD control [339], and passivity-based control [246, 3]. Most models used in the Automatic Control literature use a compliant linear spring-dashpot contact model [78] (sometimes because they lack of a suitable simulation tool), or completely neglect the dynamical effects of the impacts.

*   **Control for manipulation [19]:** Notice that the strategy : 1) guarantee force or form closure (with model-based, analytic, or data-driven approaches [359]), 2) design of feedback $\tau$, follows again a backstepping approach. "Classical" control algorithms $\tau$ may be used [19, sect. 4].

*   **Open-loop control:** this has been applied mainly to juggling systems, with so-called blind jugglers [360, 164, 278, 361]. Perturbation analyses [360] show that it may possess some local stability/robustness properties, which depend a lot on contact geometry.

## 6.7. Taylored Controllers for Control Subtasks

Another way to simplify the control problem, is to restrict the dynamics of (1) to specific motions, like walking, or running, or hopping of a biped robot, instead of considering more complex tasks. This is also typical in manipulation, where authors often tackle the problem of grasping (persistent contacts), or of
non-prehensile manipulation (contacts may be activated/deactivated). An interesting example is floating objects manipulation [77], where objects may be rigid like vessels, or deformable like liquids, and “robots” are tugboats acting on the object directly or through cables. Then various control strategies are usually employed which correspond to different goals: caging, pushing, towing. It is sometimes difficult to choose where to classify some approaches (in this section or in section 6.6?) because the dynamics, including controllability properties, and the control task may be intimately related (e.g., a one-DoF hopping robot can hardly do anything but hop vertically).

* In some important cases (balancing bipeds [31, 32], object grasping [16, 17, 18, 19, 20], kinematic chains with joint clearance [3]) it is desired to impose persistent contact, hence impacts are absent from the design. The contact framework is therefore that of section 2.1: [32] use (4) assuming that (5) holds true, hence $\lambda_n = -D_{nn}(q)^{-1}(G(q, \dot{q}, t) + \tilde{\tau})$. The algorithm in [32, P1-P6] follows a backstepping strategy.

* In the same vein, and continuing the previous item's subject, the control design is sometimes done assuming that the system remains all the time in a persistent contact mode, see e.g., [256, 255, 362, 363], which means that the input constraint (5) is always satisfied. This may be justified in some cases (if the payload of a crane system is very large so that the slackness mode is likely to never be activated and the cable remains taut, or if a frictional device is used in a frictional oscillator system [363]). Clearly in general robot-object systems this is a bold assumption. The persistent contact dynamics are used in [362, Equations (3)-(12)] to transform the dynamics in a way similar to (20), where the multipliers no longer appear. Then further equalities are used to obtain the dynamics at the object's level, as usually done in grasping (see section 2.3). The obtained transformed system has the required triangular form and is controlled with a backstepping scheme.

* Control *via* switching coordinates (a strategy largely employed for walking robots, see section 3.3.2): the switchings occur between bilaterally constrained dynamics with reduced independent coordinates. In between two switching times, the system is a classical fully or underactuated system, and any controller designed for such systems can be applied. The difficulty is to concatenate all such phases for stability analysis, incorporating impacts and input constraints. Many different control approaches exist to guarantee “stability” (roughly, the robot does not fall if its gravity center or its zero-moment-point lies in a certain location), and input constraints as a result of one sticking foot during a step. Model-predictive-control has become a popular control strategy because of its intrinsic ability to verify constraints (sliding-mode could be an alternative choice, from this point of view, as proposed in [364]). The control of the angular momentum and/or the center of mass through the reaction forces at each foot is tackled in [177].

## 6.8. *Classical Control Techniques*

As alluded to above, classical control schemes (i.e., control schemes which have been widely studied and applied to unconstrained Lagrange systems) can be applied in some instances, like in-between two events where the dynamics are that of a smooth fully or underactuated system. However the system's overall stability analysis along complex paths (26) (27) may be hard, and it is not always analysed in the literature, where many articles propose instead pure heuristics or incomplete analyses. Let us mention a few of these control algorithms (which concern the robot's input $\tau$ in (1) (b)):

* **Adaptive control:** Once a controller has been designed and shown to guarantee a desired closed-loop behaviour, classical methods to avoid perfect knowledge of parameters can be applied. This may concern “classical” on-line estimation of the robot's inertial parameters (e.g., during flight phases, or persistent
contact phases), or, more specifically for our systems, the on-line estimation of other parameters like the coefficient of restitution [15], or direct adaptive control without system's parameters estimation [365], or the on-line adaptation of the desired trajectory [197]. The on-line estimation of all inertial parameters (robot and object) has apparently not been tackled yet.

*   **Optimal control:** this can *a priori* be applied to any "classical" system with inputs and sufficient reachability (should it be continuous or discrete-time like a Poincaré mapping). The optimal control of linear complementarity systems, which has been tackled in [366, 367, 368, 369], applies to both dynamics in (55) and (57), and to all robot-object systems with compliant contact. It also applies to the circuits in (72) and (74). The dynamics in (46) can be recast into first-order sweeping processes (FOSwP) with state-dependent set [54]. The optimal control of FOSwP has received much attention recently, see [54, section 4.3]. The optimal control problem of complementarity systems with relative degree two (which yields second-order sweeping process [54, section 4.2.1]), impacts, friction and dimension variation along the solutions, is a tough objective, both from theoretical and numerical aspects (especially when constraint activation/deactivation times are assumed to be unknown). Basically this is an optimal control problem under strongly nonlinear constraints: the Lagrange dynamics, equality (bilateral) constraints, complementarity constraints (yielding MPEC problems which are nonconvex), set-valued constraints if Coulomb' friction is adopted (which can also be expressed with complementarity), and velocity jumps (which are trigerred by state conditions). However research is progressing quickly in this field, where reinforcement learning methods (which are in essence close to optimal control) may bring much [370, 371], at least from the computational point of view. Comparisons with more classical optimal control methods which are studied for nonsmooth systems [366, 367, 368, 369], with specific numerical methods [372], will be mandatory in the future. A survey on numerical optimal control for legged robots is proposed in [373].

*   **Model Predictive Control (MPC):** It has the advantage over other class of feedbacks, that constraints of equality or inequality types can be taken into account in a kind of real-time way, which facilitates the stability when several kinds of constraints are present on the input and the state. It has been applied mainly to biped walking robots [25, 331] and to quadrupeds control [374, 375]. In [375] the object's control is analyzed with the contact forces as inputs (see section 4.2.1).

*   **Sliding-mode-control (SMC):** this is attractive because of it guarantees finite-time convergence to a surface. Its main drawback is the input and output chattering, which occurs when, for instance, an inappropriate discretization method is applied. The implicit Euler method performs very well and allows to keep the nice properties of the continuous-time system, see [376] for details. Another drawback is the need, in many cases, of the output derivative to construct the sliding surface. Differentiators can be of great help, if, once again, they are correctly discretized [377, 378]. Interestingly enough, it may be, in this setting, that the finite-time convergence is more useful than the matching disturbance rejection capability (the finite-time feature is shared with dead-beat control, section 6.2). SMC has been applied to bipedal walking tasks in the pioneering work [364], and to inertially-driven robots in [189].

*   **Passivity-based Control (PBC):** it is analyzed in [31] for bipeds balance, assuming both feet perfectly stick to the ground, using the canonical form (1) [31, Equation (11)]. PBC is studied in [320] for vibro-impact robot-object systems (a subclass of (26) (27) [6, section 1.3.2]). The framework is that of object's control through pre-impact velocities (see section 4.2.1), and it extends the well-known passivity+zero-state detectability output feedback stabilization result [97, section 5.5]. The IDA-PBC method is applied in [379, 380, 381]. It uses the Hamiltonian framework. The compass model with one foot stuck to the ground
(similar to (49) in a reduced coordinate dynamics, see section 3.3.2) is used in [381]. The effects of impacts are studied numerically. The IDA-PBC is applied to the stance phase without impacts in [380]. The foot-ground contact is divided into three phases in [379], but transitions (impacts and constraints deactivation) are not taken into account. In both [380, 379] the contact multiplier is calculated assuming independent bilateral constraints, and a dynamics similar to (6) or (17) (but without taking into account constraints) is used to design the IDA-PBC input. It is noteworthy that PBC has been used in [146, 148] to facilitate the analysis at impact times, being the closed-loop's Lyapunov function close to the mechanical energy. Apparently such objective has not been tackled in the above references. PBC offers a nice framework for trajectory tracking, but this involves specific problems, see section 6.9.

* Contact force feedback (tactile control): recently this has been recognized as an important capability in some robotic tasks [215, 382, 383]. Which contact force models should be chosen, rigid or compliant ([215] extends a procedure outlined in [97, section 5.5.3] and using Lyapunov functions as in [384])? As seen above, only tactile feedback may change the structure of the CP. Problems with measurement delays specific to contact tasks are studied in [385, 386]. From a technological point of view, sensors able to detect slippage, vibrations, deformations, pressure, etc, are needed to implement such feedback inputs. Tactile control is certainly needed for some complex tasks.

~~Most (but not all) of the above classical algorithms are proposed without complete stability analysis and relying on oversimplified contact/impact models. In a sense, designing first $\tau$ in (1) boils down to follow a reverse backstepping strategy, or to consider that the object's dynamics (1) (a) are a kind of perturbation acting on (1) (b).~~

## 6.9. Trajectory Tracking

Trajectory tracking control occupies a specific place in the control of nonsmooth systems with state jumps and variable dimension, and can be tackled from different point of views, depending on whether or not the impacts are considered as disturbances [159, 161, 146, 1, 310, 148, 8, 311, 312, 313] (in which case the peaking or mismatch phenomenon is present, see section 4.4), or not [3, 339, 387, 160, 162, 159, 246]. Two main classes emerge: fully-actuated systems [159, 161, 146, 1, 310, 148, 8, 311, 312, 313], and underactuated robot-object systems. Dead-beat controllers can be used to define and stabilize desired trajectories in jugglers, in conjunction with a specific definition of desired trajectories fitted to a juggling system [334, Algorithm 1]. The object's desired trajectory is defined at impact times in [13, 14]. The desired trajectory is adapted on-line in [197] to fix its value at an impact time. The stabilization of existing trajectories when a sinusoidal excitation is applied (e.g., bouncing dimer's characteristic motions, or impact oscillators) could be performed using simple inputs, yielding at least local stability. In general, a desired trajectory has to verify the system's constraints, and the invariants (energy, momentum). This often leads to on-line modifications, or adaptations, of the desired trajectory along the solution evolution during a transient period in order to cope with the system's mode change in (26) (27). Thus the mere problem of the desired trajectory definition may not be trivial in some complementarity Lagrangian systems [146, 4, 148, 388, 312]. For systems with joint clearances, trajectories can be designed for the nonsmooth system [339], but also for the smooth system (without clearances). Then a robustness study with respect to impacts has to be led [339, 246]. In general, a reference trajectory is assigned only to an output, e.g., the center of mass' state of a biped during walking or running or jumping [331] (which is (1) (a)), and not to the whole state $(q, \dot{q})$: this stems from the
underactuation as well as from mere control objectives and stability criterion. Moreover this reference has to be adapted on-line depending on the changing environment.

The peaking phenomenon (see section 4.4) may then be a major issue for the stability of the tracking error dynamics. It may occur for instance if a classical trajectory tracking controller is applied to a walking biped during a step phase, then an impact occurs, and the controller is applied once again during the next step. Tracking for systems in billiards has been tackled in [161, 160, 162]. In all these works a great deal of effort is put on the desired trajectory definition to cope with the peaking phenomenon one way or another (this was apparently first advocated in [146]). Interestingly enough, the controlled dynamics plus the desired trajectory dynamics in [159, Equations (13) and (19)] [161, Equations (16)] [162, Equations (2) and (4)], possess a robot-object vibro-impact structure (no persistent contact).

~~On-line modification/adaptation of the desired trajectories is an ubiquitous design tool for the tracking control in complementarity Lagrangian systems, including (1).~~

## 6.10. Switching Control

Robot-object systems (1) are fundamentally switching systems, see (26) and (27). Therefore it is natural to think of their control with switching inputs:

*   **Switching-DAE model and control:** basically, complementarity systems can be seen as systems that switch between differential-algebraic equations, where the equality constraints are given by the CP modes. The switching-DAE approach has been widely used in biped locomotion control [30]. In a broader context, each equality-constrained subsystem can be controlled using techniques derived for DAE [127, 389]. However, one should not forget the fact that the switching rule is given by the CP (which is a controlled LCP), secondly switchings which involve an increase in the system's dimensions (in other words, the number of active constraints is augmented) are necessarily accompanied, in the case of (1), i.e., of relative degree 2 systems, by impacts (this is sometimes neglected [390]), thirdly it is not always easy to get an independent set of generalized coordinates that is valid globally for each DAE (this is easy for biped robots). Finally the number of DAE systems to be taken into account, increases exponentially fast with the number of unilateral constraints $m$, and friction adds modes due to stick and slip phases: this may be a strong obstacle in many cases which are not “toy systems” (as most of those presented in this article, and which are just meant to illustrate the purpose).

*   **Switching controller:** in the framework of fully actuated complementarity Lagrangian systems, switching passivity-based inputs have been analyzed in [146, 147, 148, 310]. Consider the simple example in sections 3.1.1, 3.1.2 and 3.1.3: a general task involves juggling (section 3.1.1), catching (section 3.1.3), persistent contact (section 3.1.2). Therefore switching between three “basic” controllers may be one solution. In the bipedal robots control, switching coordinates yield switching inputs at each step. But the switching can also stem from multicontact sole/ground model, which implies that (6) or (10) or (17) are changing along the foot/ground contact process: heel, flat, toes successively activate/deactivate unilateral constraints (the index set $\alpha(t)$ in (26) increases and then decreases). The passivity-based (IDA-PBC) controllers designed in [380, 379] switch along such foot/ground contact evolution (since the Delassus matrix changes), but the input constraints and the collisions are not incorporated in the study. Switching controllers seem mandatory for some clearance systems as in Figure 14 (b) [339].

*   The switching-coordinate approach that is widely used in biped walking control, see section 6.7, yields switching controllers.
## 6.11. Robustness Issues

As is well-known [391], the most-difficult-to-deal-with source of uncertainties in (1) may be the contact geometry, through $\frac{\partial h}{\partial q}$ and $H_t$. This propagates in all the developments in sections 2.1, 2.2, 2.3, through the matrices $D_{nn}(q)$, $D_{nt}(q)$, $G(q, \dot{q}, t)$, $H_c(q, \dot{q}, \tau, t)$, etc. Let $h(q) = h_o(q) + \Delta h(q)$. The (frictionless) CP in (3) becomes: $0 \le \lambda_n \perp (\nabla h_o^T M^{-1} \nabla h_o + \Delta D_{nn})\lambda_n + G + \Delta G + \tilde{\tau} + \Delta \tilde{\tau} \ge 0$, with $\Delta D_{nn} \stackrel{\Delta}{=} \nabla \Delta h^T M^{-1} \nabla \Delta h + \nabla h_o^T M^{-1} \nabla \Delta h + (\nabla h_o^T M^{-1} \nabla \Delta h)^\top$, $\Delta G \stackrel{\Delta}{=} -\nabla \Delta h^T M^{-1} F + \frac{d}{dt}(\nabla \Delta h^T) \dot{q}$. Thus, uncertainties gradients and their time-derivative should not be too large just to guarantee the CP's well-posedness: CP's ill-posedness main sources are friction and uncertain contact geometry. In manipulation uncertainties mainly stem from object's geometry, in biped robots from ground's geometry. Tools from Complementarity Theory (LCP stability, Aubin's calmness) are at the core of this topic.

# 7. Time-discretization for Simulation and Control

Some controllers (the best examples being MPC, direct method for optimal controllers, but this is also the case of SMC [376, 392, 393]) can be formulated from a time-discretization of the continuous-time plant dynamics, including all constraints, computed and analyzed this way. Then some questions arise:

* Discrete-time implementation: what is its influence on stability? Could the implicit method, which has proved to be successful in numerical simulation (see next item) and in sliding-mode control and differentiators [376, 392, 393, 377, 378], be applied safely for the implementation of robot-object systems? Consider the frictionless case, then the Euler implicit discretization of (1) yields the LCP at time $t_{k+1}$ and if $h(q_k) \le 0$ (i.e., the contact mode is activated, see the introduction of section 2.1):

$$0 \le \mu_{n,k+1} \perp D_{nn}(q_k)\mu_{n,k+1} - \nabla h(q_k)^{\top}\dot{q}_k + hG(q_k, \dot{q}_k, t_k) + h\tilde{\tau}_k \ge 0, \quad (84)$$

where $h > 0$ is the timestep (or sampling period), $\mu_{n,k+1} \stackrel{\Delta}{=} h\lambda_{n,k+1}(q_k, \dot{q}_k, t_k, \tilde{\tau}_k)$. This is the discretized version of (3), where the implicit part is in the CP discretization. The solution $\mu_{n,k+1}(q_k, \dot{q}_k, t_k, \tilde{\tau}_k)$ to the LCP (84) is piecewise linear in $\tilde{\tau}_k$ and it can be inserted in a discretized version of (1) (for instance an explicit Euler discretization for the smooth terms), to get a piecewise continuous discrete-time system. For (1) the Euler discretization reads as:

$$\left\{
\begin{array}{l}
(a) \quad M_1(q_{1,k}) \dot{q}_{1,k+1} = M_1(q_{1,k}) \dot{q}_{1,k} - hF_1(q_{1,k}, \dot{q}_{1,k}, t_k) + \frac{\partial h}{\partial q_1}^T (q_{1,k}) \mu_{n,k+1}(q_k, \dot{q}_k, t_k, \tilde{\tau}_k) \\
(b) \quad M_2(q_{1,k}) \dot{q}_{2,k+1} = M_2(q_{1,k}) \dot{q}_{2,k} - hF_2(q_{2,k}, \dot{q}_{2,k}, t_k) + \frac{\partial h}{\partial q_2}^T (q_{2,k}) \mu_{n,k+1}(q_k, \dot{q}_k, t_k, \tilde{\tau}_k) + hE(q_{2,k})\tau_k \\
(c) \quad 0 \le \mu_{n,k+1} \perp h(q_{1,k}, q_{2,k}) \ge 0 \\
(d) \quad \text{Impact and friction models,}
\end{array}
\right.
\quad (85)$$

with $q_{1,k+1} = q_{1,k} + h\dot{q}_{1,k+1}$, $q_{2,k+1} = q_{2,k} + h\dot{q}_{2,k+1}$. The algorithm in (84) (resp. (85) (a) (b)) is the discrete-time counterpart of (3) (resp. (6)). The advantage is that one can formulate discrete-time problems for control design (optimal control or else), which can be solved with efficient numerical solvers. Most importantly, when embedded into Moreau's algorithm (see (86)) this discretization encompasses impact times, because the primary variable in the algorithm is $\mu_{n,k+1}$ [7], which is always bounded. The control of the discrete-time system (84) (85), PWL in $\tau_k$, can be studied for each $h > 0$. The application of the discrete-time controller to the continuous-time system (1), then deserves close attention: how does the stability in the discrete setting evolve when $h > 0$? Is the closed-loop system well-behaved when $h > 0$?
becomes small ? Also the discrete-time system (85) can be used in a dynamic programming environment to compute some optimal trajectories. The presence of complementarity constraints yields a particular class of optimization problems, named MPLCC (mathematical program with linear complementarity constraints) or MPEC (mathematical program with equilibrium constraints). At this stage a discussion about (85) and its relationship with (78) is mandatory. In essence, (85) is an event-driven method [7], i.e., one needs to apply some event-detection algorithm to detect impacts and stick/slip transitions. Another, powerful approach consists in discretizing (78) with an event-capturing time-stepping scheme (named the catching-up algorithm by J.J. Moreau [394, 55]):

$$ \left\{ \begin{array}{l} M(q_k)(v_{k+1} - v_k) + hF(q_k, v_{k+1}) - \begin{pmatrix} 0 \\ hE(q_{2,k})\tau_k \end{pmatrix} \in -\mathcal{N}_{\mathcal{T}_{\Phi}^h}(q_k) \left( \frac{v_{k+1} + e_n v_k}{1 + e_n} \right) \\ q_{k+1} = q_k + h v_{k+1} \end{array} \right. \quad (86) $$

which is the same kind of scheme as (84) (85), with a specific impact model. This algorithm is said to be driven by the velocity, because once a penetration inside the admissible domain $\Phi$ is detected, it takes one step to reverse the normal velocity. It is implicit because $v_{k+1}$ is the argument of the set-valued normal cone. A great deal of work has been published about (86) and its variants, see [7]. The Moreau-Jean event-capturing algorithm, which is an extension of (86) with a $\theta$-method [55, 395, 396, 6, 7] allows to solve numerically generalized equations as (19), with frictional contact. Specific event-capturing schemes inspired from Moreau-Jean are used in [397] to compute optimal controllers (with an application on a jumping robot in [368]). Another, less known, time-stepping scheme is the Schatzman-Paoli algorithm [398, 399]. The main discrepancy with respect to (86) is the set-valued right-hand side which is $-\mathcal{N}_{\Phi} \left( \frac{q_{k+1} + e_n q_{k-1}}{1 + e_n} \right)$: thus it is a position-driven scheme. The reader is referred to the original articles for details on the required assumptions on $\Phi$ and convergence results for both Moreau-Jean and Schatzman-Paoli event-capturing time-stepping algorithms.

For control design and implementation purposes, the above compact forms are interesting because they retain some properties like maximal monotonicity which is a form of incremental passivity [97, p.295]. However for simulation purposes, they need to be transformed to tractable forms like LCP.

* Implicit vs explicit discretizations: it is a well-known fact that implicit methods add dissipation (numerical dissipation), while explicit methods decrease dissipation. Semi-implicit methods (like the $\theta$-method), and others, may possess interesting properties like preservation of some quantities (energy, momentum, or other dynamical invariants). In Control design both the plant and the controller are discretized. How much these discretizations influence the closed-loop stability for positive sampling periods would be worth investigating.

* Numerical simulation (implicit event-capturing time-stepping methods): the Moreau-Jean algorithm [395, 396] and its variants (like the Stewart-Trinkle algorithm[400]) prove to be very efficient for systems with unilateral contact, impacts and set-valued friction [7, 401]. Alternative approaches [402] use compliant contact models of class c) in a complementarity framework (see sections 2.2 and 3.6.1) and propose time-stepping $\theta$-schemes, keeping Coulomb's friction set-valuedness (hence a correct simulation of sticking modes). To the best of the author's knowledge, commercial software toolboxes do not incorporate such time-stepping schemes (or, they are based on regularized friction, that is to be avoided to take sticking modes into account). Several commercial or open-source codes have been developed in the past years. The open-source INRIA
library SICONOS [401]<fnref n="3" /> is dedicated to nonsmooth systems (mechanical or circuits). It is mainly based on the Moreau-Jean event-capturing time-stepping scheme, but other complementarity systems like LCS can be simulated. Some Robotics simulators incorporate rigid contact models [123], while others advocate the use of compliant contact/impact models [402]. To conclude, the choice of the contact/impact model is still the object of research and debate in the Robotics scientific community, but recent contributions seem to indicate that Euler-like schemes could be the most efficient ones to handle (1) when *m* is large and friction is present. It is noteworthy that event-capturing schemes may be used for parameter and contact forces estimation [403]. It is possible that other types of contact/impact models have to be used to simulate some systems. Let us note that the Darboux-Keller shock dynamics (which belong to impact models of class b), see section 2.2) can also be discretized with a time-stepping scheme [6, Proposition 5.29].

## 8. Further Perspectives

Let us provide an attempt about possible future research directions, in addition to those mentioned at several places in the article:

1. A better understanding of common features of the subclasses of robot-object systems presented in this article is needed: as we saw, in spite of the fact that they all share the same global structure that lends itself naturally to a backstepping strategy, it remains unclear whether or not they could be recast into a small number of subclasses sharing same control properties. This has been done in some cases (e.g., jugglers ≈ hoppers) but a more systematic analysis is still lacking. A first direction of research could consist of analyzing each one of the above examples in detail.

2. Notions and parameters that can be used for several subclasses and tasks:

(a) form or force closure for all robotic tasks which involve sticking contacts (balancing bipeds, manipulation, climbing machines): in all cases the challenge is to determine conditions under which contacts are kept and can counteract disturbances. This involves the sticking contact problems detailed in section 2.1. An underlying question concerns the similarities/differences between the climbing robot in Fig. 13, and the prehensile manipulation robot in Fig. 9. It has also long been understood that stability notions developed primarily for manipulation systems [404, 288, 405] also apply to biped robots stability analysis [56].

(b) kinetic angles between the unilateral constraints [6, p.403]: kinetic angles encapsulate both geometrical and inertial effects, hence their great usefulness as a parameter for (1) (the Delassus' matrix $D_{nn}(q)$ in (2) is the matrix of kinetic angles: it is diagonal if all constraints are mutually orthogonal in the kinetic metric). Their influence is obvious in certain cases (stability: slenderness of rocking blocks, gait length in a walking biped robot for a given masses distribution; catching task with obtuse or acute angle), where the discontinuity with respect to initial data is an important property. They also play a role in impact controllability [295, 1]. Their role is important in other applications (dimers on a plate have the ability to move forward/backward because of their kinetic angle and friction). This is sometimes called the aspect ratio in the rocking block studies. A new kind of kinetic angles, defined from the Delassus' matrix, is introduced in section 4.2. This seems to play a crucial role in the object's controllability from $\tau$ during persistent contact modes.

<footnote n="3">³https://nonsmooth.gricad-pages.univ-grenoble-alpes.fr/siconos/index.html</footnote>
(c) number of constraints $m$: multiple unilateral constraints can be used in a sequential way (sequential impacts, contact-switching between various surfaces with constraints activation or deactivation), or in a simultaneous way (multiple impacts, or persistent contact with several surfaces). It is clear that $m$ plays a major role in legged locomotion (a quadruped is usually easier to stabilize than a biped), object manipulation (the more fingers the more manipulability). But, kinetic angles also may play a significant role (see the previous item). In some applications $m$ can be very large (e.g., a biped's foot with detailed sole modeling).

(d) Poincaré mappings: find applications in all systems with repeated-impact trajectories (vibro-impact systems, jugglers, running bipeds) or with repeated stick-slip transitions. The Poincaré surfaces are usually chosen from (26) and (27).

3. Impact modeling: the models introduced in section 2 could a priori be applied to any complementarity Lagrangian system. But, depending on the system, on the type of study (controllability, control, simulation) and on the envisaged task, different models may have to be used (e.g., impact models of classes a), b) or c)). This has been studied in Mechanical Engineering where comparisons/validations have been performed for various systems like granular chains [106, 110] or planar rocking blocks [406, 407]. The development of taylored contact/impact models may be needed for some complex systems/tasks. For instance bipeds that walk/run/hop may have complex behaviour at feet/ground impacts. The energetics of the system incorporating impacts is an important feature [408], apparently not so much analysed in the Robotics literature.

4. Unified control frameworks:

(a) As pointed out in the foregoing sections, backstepping furnishes a generic three-step framework for the control design of (1), no matter the control objective may be (impacting trajectories, persistent contact trajectories, or both). This is intimately linked with the interaction potential function between the robot and the object (see section 1). How do the particular features of robotic tasks influence the control design at each step of the backstepping algorithm? This is certainly a complex problem, and only partial answers have been given so far.

(b) Several robot-object systems are equivalent (jugglers and hoppers, three-ball and particle hitting an angle, etc). Control strategies which apply to several subclasses of (1) have been proposed in [1] (recursive backstepping strategy, impacting trajectories, dead-beat control of the robot), [15] (similar, with adaptive control for the restitution, one-DoF hoppers and jugglers), [3] (persistent-contact control and impacting trajectories, kinematic chains with clearance), [70] (use the analogy between bipedal locomotion and some manipulation mechanisms), [31] (persistent-contact in biped balancing and grasping tasks), [298] (combined impulsive and continuous-time inputs to stabilize hopping, tapping, crawling, walking simple systems), [199] (stabilization of various gaits -tapping, galloping, hopping- for an inertially driven system). Is there a unique control strategy that could be designed for (1), encompassing all possible robotic tasks? Or is this just out of reach, so that we should better look for subclasses (as large as possible) of (1) which enjoy sufficient common features to be embedded in a single control strategy (itself embedded in a backstepping algorithm)? Or should we look for both subclasses of (1) and associated specific tasks (bipeds which run, or which walk; machines which juggle, or which hold firmly objects)? As alluded to above, future robotic tasks may be more and more complex, so that robots may have to perform tasks which involve all phases of motion, and classifying them into subclasses may lose
its usefulness and meaning: a legged machine will have to walk, to run, to hop, to stay balanced, with sticking or slipping feet, while taking and putting different objects with the hands. Even if switching between several low-level feedback controllers may be planned, the overall behaviour of (1) under such nonsmooth controllers has to be studied. Due to the switching nature of (1), controllers which enjoy finite-time convergence properties and which accomodate with inequality constraints (siding-mode and model-predictive controllers have long been known to be suitable in this respect), seem to be good candidates. MPC has been applied to bipedal robots [25] and to juggling systems [348].

(c) Underactuated robots: the robot in (1) (b) can be underactuated while the control objectives are still attained, see [237] for the case of a snake robot, [409, 141] for juggling systems. It could be of interest to determine the role of the robot-object interactions in the ability of the system to bear a higher degree of underactuation. For instance, anisotropic friction seems to be quite useful to improve the set of possible trajectories. The control of systems with very high underactuation degree $n_1 + n_2 - p$ (i.e., $n_1 \gg 1$) as in Fig. 3 (a), 4 (b) and 5 (c), remains largely open. This certainly implies to consider new controllability notions and new control targets.

(d) As recalled above, the dynamics (1) are efficiently time-discretized with event-capturing time-stepping schemes. Could the control design be analyzed in discrete-time from these schemes, which have the advantage of using constant time-steps (or sampling periods in a control-oriented language), and including all phases of motion, including impacts and dimension changes ? Preliminary promising results are in [307, 345].

5. State observers could be developed for (1) with output $(q_2, \dot{q}_2)$, to estimate $(q_1, \dot{q}_1)$. Here it seems that two main strategies could be planned: observability through impacts [114, 306], and observability with persistent contact (then a specific input has to be applied to keep the contact, see section 2.1). Due to the (usually) finite duration of modes, finite-time observers and differentiators could be quite useful (then it is crucial that their digital implementation be realized correctly, see [377, 378]).

6. Data-driven (empirical, machine learning, AI techniques, big data) approaches: from a general viewpoint, it is known that Automatic Control with “classical” modeling and Machine-Learning are complementary one each other [410, 411, 9]. Data-driven methods have been applied in manipulation and grasping [359, 75, 166, 412, 382, 143], biped walking machines [413, 414, 415, 416], hoppers [415], quadruped machines [134], juggling systems [417, 418, 419, 420]. They may be model-free methods, or incorporate some modeling (like the restitution model [417], Lagrangian dynamics [419]), with or without feedback (reward functions are a feedback action which uses the error between the actual and the desired trajectories [417]). As noted in [417], applying reinforcement learning to a system as simple as a ping-pong robot player (similar to the juggling system in Fig. 3 (b)), would require millions of exploring and failing tasks, which is impossible. Thus in general training a robot for all possible scenarios is just out of reach due to necessary experimental time, and high cost [415]. Consequently it is likely that these approaches do not apply uniformly to all systems (1), depending on parameters $n, m, \bar{m}$, CP uncertainties, etc, on the control objective and task complexity. One solution is the use of good, reliable simulators to replace the experiments [415]. Complementarity and input constraints usually play significant roles in nonsmooth systems, hence incorporating good contact models is of utmost importance. As noted above, very few (open-source or commercial) software packages are dedicated to the simulation of nonsmooth mechanical systems [7, 401]. Though these methods intrinsically possess
some universal (model-free) properties, our nonsmooth systems still pose interesting challenges [421]. But, as alluded to in section 6.8, reinforcement learning and optimal control are likely to feed one each other [370, 371] (probabilistic model predictive control together with probabilistic model-based reinforcement learning techniques [416]), making possible the synthesis of highly complex controllers in a near future. Friction models can be derived from data-driven algorithms [102]. An important point is to determine to which accuracy data-driven methods approximate the complex dynamics (1) so that it does not deteriorate too much the performance, to which type of task they apply ([422] points out serious difficulties with too stiff contact, [419] incorporate unilateral constraints in their model-based reinforcement learning algorithm), and how much effort is necessary to implement them (data-driven methods being far from some kind of universal solution to all problems [410]). Koopman operators method, which consists of linearizing a finite-dimensional system by replacing it with an infinite-dimensional operator [423, 424] has been applied to some hybrid systems [424, 425]. It remains to be proved that it applies to complementarity systems, using for instance the discrete-time event-capturing schemes introduced above.

7. Deformable bodies: the manipulation of deformable objects with rigid-body robots [426, 427, 216, 262], of rigid objects with flexible arms (grasping with beams) [428], of rigid objects with sheets or clothe-like robots [76], flexible inchworm systems [429], require models that stem from continuum mechanics. There certainly still exist many open challenges for their control as robot-object systems modeled with coupled ODE and PDE, especially when impacts and friction are involved.

8. Landing aircrafts on moving platforms: as recalled in section 3.1.8 most of the results in this field do not consider possible impacts, hence may lack of robustness. Consider Fig. 21, an interesting issue is the stabilization of $m_2$ (the landing aircraft) on $m_1$ (the ship or the platform on the sea) when $m_1$ is connected to the wall with a linear or nonlinear spring-dashpot system [6, chapter 2]. Its dynamics are given by:

$$ egin{cases} (a) \ m_1\ddot{q}_1 + k_1(q_1 - l) + m_1g = -\lambda_n \\ (b) \ m_2\ddot{q}_2 + m_2g = \lambda_n + \tau \\ 0 \le \lambda_n \perp h(q) = q_2 - q_1 - R \ge 0. \end{cases} \quad (87) $$

According to our terminology, (87) (a) is the object, (87) (b) is the controlled robot. In a sense this is a reversed juggling machine.

<figure><img src="image_21.png" /><figcaption>Figure 21: Aircarft landing on a ship.</figcaption></figure>

9. Passive control of frictional oscillators with energy sinks using nonlinear stiffnesses is a well-studied topic in Nonlinear Dynamics, see [430] and references therein. Could it be a source of inspiration for the design of nonlinear dynamic controllers for this type of robot-object systems ? For instance [430, Equations (3) and (9)] can be seen as a dynamic state feedback with nonlinear PD. The objective is to reduce vibrations due to friction. Similar goal could be targetted with impact energy absorbers to study, e.g., gear rattle noise decrease. Can active control compete with passive control in such
## 9. Conclusions

The objective of this article is mainly to show that the robot-object class of nonsmooth underactuated mechanical systems, gathers many different mechanical systems which share the same global structure. These systems have the peculiar feature that contacts, impacts and/or friction are necessary to achieve their control: they are contact/impact-aware systems. Firstly we present the main analytical tools which are mandatory using for the understanding of their dynamics, as well as some modeling ingredients which are well-suited to such multibody dynamical systems: the contact complementarity problem with or without friction, impact and friction models, some dynamics transformation, fixed-points calculation. Secondly a list of various systems is provided which all can be recast into the robot-object class. Most of the examples are mechanical systems, but it is shown that some circuits (electrical or hydraulics) can also be recast into robot-object systems. In each case the dynamics are written in detail and some comments are made. Thirdly the main contributions in the Automatic Control and the Robotics scientific literatures concerning robot-object systems' control are summarized: the analytical tools for control (stability, controllability, passivity, etc), the main control strategies, and some modeling aspects. It appears clearly that in spite of the fact that several subclasses of robot-object systems have been studied thoroughly in a separate way, a deep reflection about how to use their similarities (when they exist) for their control is still largely lacking. It is advocated in this article that the backstepping framework could furnish a useful framework to that aim.

## Appendix A. Lagrange Dynamics of the Jumping Robot in (49)

The coordinates of $O_1$ and $O_2$ in $(O, i, j)$ are $(x_1, y_1)$ and $(x_2, y_2)$, respectively. The kinetic energy is $T = \frac{1}{2}m_1(\dot{x}_1^2 + \dot{y}_1^2) + \frac{1}{2}m_2(\dot{x}_2^2 + \dot{y}_2^2)$. The following relations hold:

$$ 
\begin{aligned}
x_1 &= x_c + l_1 \sin(\theta_1), \quad y_1 = y_c - l_1 \cos(\theta_1), \quad x_2 = x_c + l_2 \cos(\theta_2), \quad y_2 = y_c + l_2 \sin(\theta_2), \\
x &= \frac{1}{m_1+m_2}(m_1x_1 + m_2x_2), \quad y = \frac{1}{m_1+m_2}(m_1y_1 + m_2y_2), \\
x_1 &= x + \frac{m_2}{m_1+m_2}(l_1 \sin(\theta_1) - l_2 \cos(\theta_2)), \quad y_1 = y + \frac{m_2}{m_1+m_2}(-l_1 \cos(\theta_1) - l_2 \sin(\theta_2)), \\
x_2 &= x + \frac{m_1}{m_1+m_2}(-l_1 \sin(\theta_1) + l_2 \cos(\theta_2)), \quad y_2 = y + \frac{m_1}{m_1+m_2}(l_1 \cos(\theta_1) + l_2 \sin(\theta_2)), \\
\dot{x}_1 &= \dot{x} + \frac{m_2}{m_1+m_2}(l_1 \dot{\theta}_1 \cos(\theta_1) + l_2 \dot{\theta}_2 \sin(\theta_2)), \quad \dot{y}_1 = \dot{y} + \frac{m_2}{m_1+m_2}(l_1 \dot{\theta}_1 \sin(\theta_1) - l_2 \dot{\theta}_2 \cos(\theta_2)), \\
\dot{x}_2 &= \dot{x} + \frac{m_1}{m_1+m_2}(-l_1 \dot{\theta}_1 \cos(\theta_1) - l_2 \dot{\theta}_2 \sin(\theta_2)), \quad \dot{y}_2 = \dot{y} + \frac{m_1}{m_1+m_2}(-l_1 \dot{\theta}_1 \sin(\theta_1) + l_2 \dot{\theta}_2 \cos(\theta_2)).
\end{aligned}
\quad (A.1) $$

The kinetic energy in the coordinate $q = (x, y, \theta_1, \theta_2)^\top$ is equal to:

$$ T = \frac{1}{2}(m_1 + m_2)(\dot{x}^2 + \dot{y}^2) + \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (l_1^2 \dot{\theta}_1^2 + l_2^2 \dot{\theta}_2^2 + l_1 l_2 \dot{\theta}_1 \dot{\theta}_2 \sin(\theta_2 - \theta_1)). \quad (A.2) $$

The potential energy is equal to:

$$ U = -m_1 gy_1 - m_2 gy_2 = -(m_1 + m_2)y. \quad (A.3) $$

The Lagrange dynamics $\frac{d}{dt} \frac{\partial (T-U)}{\partial \dot{q}} - \frac{\partial (T-U)}{\partial q} = Q$ furnish the mass matrix

$$ M(q) = \begin{pmatrix}
m_1 + m_2 & 0 & 0 & 0 \\
0 & m_1 + m_2 & 0 & 0 \\
0 & 0 & \frac{m_1 m_2 l_1^2}{m_1 + m_2} & \frac{m_1 m_2 l_1 l_2 \sin(\theta_2 - \theta_1)}{m_1 + m_2} \\
0 & 0 & \frac{m_1 m_2 l_1 l_2 \sin(\theta_2 - \theta_1)}{m_1 + m_2} & \frac{m_1 m_2 l_2^2}{m_1 + m_2}
\end{pmatrix} $$
and the centrifugal/Coriolis generalised forces equal to $C(q, \dot{q})\dot{q}$, where

$$C(q, \dot{q}) = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{m_1 m_2 l_1 l_2 \cos(\theta_2 - \theta_1)}{m_1 + m_2} \dot{\theta}_2 \\ 0 & -\frac{m_1 m_2 l_1 l_2 \cos(\theta_2 - \theta_1)}{m_1 + m_2} \dot{\theta}_1 & 0 \end{pmatrix}$$

so that $\dot{M}(q, \dot{q}) - 2C(q, \dot{q})$ is skew-symmetric [97, Lemma 6.17]. The angular momenta of links 1 and 2 with respect to G in the frame $(O, i, j)$ are $H_{O_1/G} = GO_1 \times m_1 \frac{d}{dt} GO_1$ and $H_{O_2/G} = GO_2 + m_2 \frac{d}{dt} GO_2$. Calculations

using $GO_i = \begin{pmatrix} x_i - x \\ y_i - y \\ 0 \end{pmatrix}$ and the relations in (A.1) yield $H_{O_1/G} = \frac{m_1 m_2^2}{(m_1+m_2)^2} (l_1^2 \dot{\theta}_1 + l_2^2 \dot{\theta}_2 + l_1 l_2 (\dot{\theta}_1 + \dot{\theta}_2) \sin(\theta_2 - \theta_1))$ and $H_{O_2/G} = \frac{m_1^2 m_2}{(m_1+m_2)^2} (l_1^2 \dot{\theta}_1 + l_2^2 \dot{\theta}_2 + l_1 l_2 (\dot{\theta}_1 + \dot{\theta}_2) \sin(\theta_2 - \theta_1))$. Hence $m(t) = H_{O_1/G}(t) + H_{O_2/G}(t)$.

The matrices $H_1(q)$ and $H_2(q)$ in (1) are calculated noticing that $\lambda_{t,1}$ performs work on $\delta x_{P_1}$, and $\lambda_{t,2}$ performs work on $\delta x_{P_2}$, with $\dot{x}_{P_1} = \dot{x} + (l_1+l_3 - \frac{m_1 l_1}{m_1+m_2}) \dot{\theta}_1 \cos(\theta_1) + \frac{m_2 l_2}{m_1+m_2} \dot{\theta}_2 \sin(\theta_2)$, $\dot{x}_{P_2} = \dot{x} - \frac{m_1 l_1}{m_1+m_2} \dot{\theta}_1 \cos(\theta_1) - (l_2+l_4 - \frac{m_2 l_2}{m_1+m_2}) \dot{\theta}_2 \sin(\theta_2)$. It is possible to add a torsional elasticity $k_{tor} > 0$ and viscous Rayleigh dissipation with coefficient $c > 0$ at the joint C. They take the form of torques in the third and fourth lines of (49) whose left-hand sides take the form:

$$\left\{ \begin{array}{l} \frac{m_1 m_2 l_1^2}{m_1 + m_2} \ddot{\theta}_1 + \frac{m_1 m_2 l_1 l_2}{m_1 + m_2} \sin(\theta_2 - \theta_1) \ddot{\theta}_2 + \frac{m_1 m_2 l_1 l_2}{m_1 + m_2} \cos(\theta_2 - \theta_1) \dot{\theta}_2^2 + k_{tor}(\theta_1 - \theta_2) + c(\dot{\theta}_1 - \dot{\theta}_2) \\ \\ \frac{m_1 m_2 l_1 l_2}{m_1 + m_2} \sin(\theta_2 - \theta_1) \ddot{\theta}_1 + \frac{m_1 m_2 l_2^2}{m_1 + m_2} \ddot{\theta}_2 - \frac{m_1 m_2 l_1 l_2}{m_1 + m_2} \cos(\theta_2 - \theta_1) \dot{\theta}_1^2 + k_{tor}(\theta_2 - \theta_1) + c(\dot{\theta}_2 - \dot{\theta}_1). \end{array} \right. \qquad (A.4)$$

These are internal torques which do not influence the angular momentum $m(t)$ conservation.

## Appendix B. Lagrange Dynamics of the Ringing Bell in (52)

The kinetic energy is $T = \frac{1}{2} m_1 V_{G_1}^\top V_{G_1} + \frac{1}{2} m_2 V_{G_2}^\top V_{G_2}$, where velocities are expressed in the Galilean frame $(O, i, j)$. Using $OG_1 = \begin{pmatrix} l_1 \cos(\theta_1) \\ l_1 \sin(\theta_1) \end{pmatrix}$, and $OG_2 = \begin{pmatrix} l_3 \cos(\theta_1) + l_2 \cos(\theta_2) \\ l_3 \sin(\theta_1) + l_2 \sin(\theta_2) \end{pmatrix}$, it follows that:

$$T(\theta_1, \theta_2, \dot{\theta}_1, \dot{\theta}_2) = \frac{1}{2}(m_1 l_1^2 + m_2 l_3^2) \dot{\theta}_1^2 + \frac{1}{2} m_2 l_2^2 \dot{\theta}_2^2 + m_2 l_2 l_3 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2). \qquad (B.1)$$

The gravity potential energy is $U(q) = -m_1 l_1 \cos(\theta_1) g - m_2 (l_1 \cos(\theta_1) + l_2 \cos(\theta_2)) g$. The torque $\tau_1$ works on $\delta\theta_1$, while $\tau_2$ works on $\delta(\theta_1 - \theta_2)$. The gap function $h_1(q)$ may be obtained by solving $\min_l ||PG_2||^2$, where $l$ denotes the distance between the bodie's upright corner and a point P lying on the bodie's right face. Denoting $||OO_1|| = l_5$, one finds the optimal $l = l_3 - l_5 + l_2 \cos(\theta_1 - \theta_2)$. Then $h_1(q) = (PG_2)^\top n$ [6, section 4.1.3], where $n = \begin{pmatrix} \sin(\theta_1) \\ -\cos(\theta_1) \end{pmatrix}$ is the inwards normal vector to the right face. The same procedure is employed to compute $h_2(q)$. The angular momentum of the ringing bell with respect to O, $H_{S/O} = H_{S_1/O} + H_{S_2/O} = m_1 OG_1 \times \frac{d}{dt}(OG_1) + m_2 OG_2 \times \frac{d}{dt}(OG_2) = (m_1 l_1^2 + m_2 l_3^2) \dot{\theta}_1 + m_2 l_2^2 \dot{\theta}_2 + m_2 l_2 l_3 (\dot{\theta}_1 + \dot{\theta}_2) \cos(\theta_1 - \theta_2)$.

## Appendix C. Lagrange Dynamics of the Systems in Fig. 20 (b) and (c)

Consider the system in Fig. 20 (b). The moment of inertia of the “body” is $I_1$, that of the disk is $I_2$, the radius at which the mass $m_2$ is placed is $r_2$, the mass of the disk is $m_3$, the mass of the “body” is $m_1$,
the “body”'s length is $l_1$, its center of mass $G_1$ is placed at distance $r_1$ from the bottom revolute joint. The kinetic energy is $T(q, \dot{q}) = \frac{1}{2}I_1\dot{q}_1^2 + \frac{1}{2}I_2\dot{q}_2^2 + \frac{1}{2}m_2(l_1^2\dot{q}_1^2 + r_2^2\dot{q}_2^2 + l_1r_2\cos(\theta_1 - \theta_2)\dot{q}_1\dot{q}_2)$. The gravity potential energy is $U(q) = r_1\sin(\theta_1)m_1g + l_1\sin(\theta_1)m_3g + m_2(l_1\sin(\theta_1) + r_2\sin(\theta_2))g$. In the minimal coordinate $q = (\theta_1, \theta_2)^\top$, the dynamics are:

$$ \left\{ \begin{array}{l} (a) \ (I_1 + l_1^2 m_2) \ddot{\theta}_1 + m_2 l_1 r_2 \cos(\theta_1 - \theta_2) \ddot{\theta}_2 + m_2 l_1 r_2 \sin(\theta_1 - \theta_2) \dot{\theta}_2^2 + (r_1 m_1 + l_1 m_3 + m_2 l_1) g \cos(\theta_1) = 0 \\ (b) \ I_2' \ddot{\theta}_2 + m_2 l_1 r_2 \cos(\theta_1 - \theta_2) \ddot{\theta}_1 - m_2 l_1 r_2 \sin(\theta_1 - \theta_2) \dot{\theta}_1^2 + m_2 r_2 g \cos(\theta_2) = \tau, \end{array} \right. \qquad (C.1) $$

where $I_2' = I_2 + (m_2 + m_3)l_1^2 + m_2 r_2^2$. Clearly $r_2 = 0$ suppresses all couplings between (C.1) (a) and (b), making it uncontrollable. The approach in [176] can be applied to (C.1), and we will see below that an extension can also apply to the nonminimal system with multipliers. Let us now pass to the system in Fig. 20 (c). We denote $OG_1 = (x_1, y_1)^\top$. The controller $\tau$ works on $\delta\theta_2 - \delta\theta_1$. The dynamics with coordinate $q = (\theta_1, \theta_2, x_1, y_1)^\top$ are given by:

$$ \left\{ \begin{array}{l} (a) \ (I_1 + (m_2 + m_3)(l_1 - r_1)^2)\ddot{\theta}_1 - (m_2 + m_3)(l_1 - r_1)\sin(\theta_1)\ddot{x}_1 + (m_2 + m_3)(l_1 - r_1)\cos(\theta_1)\ddot{y}_1 \\ \qquad + m_2 r_2(l_1 - r_1)\cos(\theta_1 - \theta_2)\ddot{\theta}_2 + m_2 r_2(l_1 - r_1)\sin(\theta_1 - \theta_2)\dot{\theta}_2^2 + ((m_2 + m_3)g(l_1 - r_1)\cos(\theta_1) \\ \qquad = -\tau - r_1 \cos(\theta_1)\lambda_{n,1} + r_1 \sin(\theta_1)\lambda_t, \\[1.5em] (b) \ (I_2 + m_2 r_2^2)\ddot{\theta}_2 - m_2 r_2 \sin(\theta_2)\ddot{x}_1 + m_2 r_2 \cos(\theta_2)\ddot{y}_1 + m_2 r_2(l_1 - r_1)\cos(\theta_1 - \theta_2)\ddot{\theta}_1 \\ \qquad - m_2 r_2(l_1 - r_1)\sin(\theta_1 - \theta_2)\dot{\theta}_1^2 + m_2 r_2 g \sin(\theta_2) = \tau, \\[1.5em] (c) \ (m_1 + m_2 + m_3)\ddot{x}_1 - (m_2 + m_3)(l_1 - r_1)\sin(\theta_1)\ddot{\theta}_1 - m_2 r_2 \sin(\theta_2)\ddot{\theta}_2 \\ \qquad - (m_2 + m_3)(l_1 - r_1)\cos(\theta_1)\dot{\theta}_1^2 - m_2 r_2 \cos(\theta_2)\dot{\theta}_2^2 = \lambda_t, \\[1.5em] (d) \ (m_1 + m_2 + m_3)\ddot{y}_1 + (m_2 + m_3)(l_1 - r_1)\cos(\theta_1)\ddot{\theta}_1 + m_2 r_2 \cos(\theta_2)\ddot{\theta}_2 - (m_2 + m_3)(l_1 - r_1)\sin(\theta_1)\dot{\theta}_1^2 \\ \qquad - m_2 r_2 \sin(\theta_2)\dot{\theta}_2^2 + (m_1 + m_2 + m_3)g = \lambda_n, \\[1.5em] (e) \ 0 \le \lambda_n \perp y_1 - r_1 \sin(\theta_1) \ge 0, \\[1.5em] (f) \ \lambda_t \in -\mu\lambda_n \operatorname{sgn}(v_t), \ v_t = \dot{x}_1 + r_1 \sin(\theta_1)\dot{\theta}_1 \\[1.5em] (g) \ \text{Impact model.} \end{array} \right. \qquad (C.2) $$

The dynamics in (C.2) (c) (d) are not an object dynamics, though they do not depend on $\tau$. The center of mass $G$ dynamics are obtained from (C.2) (c) (d):

$$ \left\{ \begin{array}{l} (m_1 + m_2 + m_3)\ddot{x}_G = \lambda_t \\ (m_1 + m_2 + m_3)\ddot{y}_G = -(m_1 + m_2 + m_3)g + \lambda_n, \end{array} \right. \qquad (C.3) $$

noting that $x_G = (m_1 + m_2 + m_3)x_1 + (m_2 + m_3)(l_1 - r_1)\cos(\theta_1) + m_2 r_2 \cos(\theta_1)$ and $y_G = (m_1 + m_2 + m_3)y_1 + (m_2 + m_3)(l_1 - r_1)\sin(\theta_1) + m_2 r_2 \sin(\theta_1)$. Replacing $\ddot{x}_1$ and $\ddot{y}_1$ in (C.2) (a) (b) by their expressions obtained from (C.2) (c) and (d), allows one to recast (C.2) into (1): the object dynamics are (C.3) with $q_1 = (x_G, y_G)^\top$, the underactuated robot's dynamics are the transformed (C.2) (a) (b) with $q_2 = (\theta_1, \theta_2)^\top$. The obtained robot's dynamics are not a Lagrangian system, however. Recasting them within (82), we have $M_{11} = M_1(q_1) = \operatorname{diag}(m_1 + m_2 + m_3)$, $M_{12}, M_{22}, F_{z_1}, F_{z_2}$ depend only on $z_2, \dot{z}_2, F_1$ is constant. Hence the system in (83) (b) is a robot's dynamics. It is noteworthy that such a transformation is similar to the one proposed in [176]. Let us note also that the obtained dynamics share several common features with (49) (like robot's dynamics underactuation), and it would be interesting to analyze them more deeply. Let us come back to the system in Fig. 20 (b). The revolute joint at the bottom tip creates the bilateral constraints
$f_1(q) = y_1 - r_1 \sin(\theta_1) = 0$ and $f_2(q) = x_1 - r_1 \cos(\theta_1) = 0$. This gives the dynamics:

$$
\left\{
\begin{aligned}
& (a) \left(I_1 + (m_2 + m_3)(l_1 - r_1)^2\right)\ddot{\theta}_1 - (m_2 + m_3)(l_1 - r_1)\sin(\theta_1)\ddot{x}_1 + (m_2 + m_3)(l_1 - r_1)\cos(\theta_1)\ddot{y}_1 \\
& + m_2r_2(l_1 - r_1)\cos(\theta_1 - \theta_2)\ddot{\theta}_2 + m_2r_2(l_1 - r_1)\sin(\theta_1 - \theta_2)\dot{\theta}_2^2 + ((m_2 + m_3)g(l_1 - r_1)\cos(\theta_1) \\
& = -\tau - r_1 \cos(\theta_1)\lambda_{n,1}^b + r_1 \sin(\theta_1)\lambda_{n,2}^b, \\[1.5em]
& (b) \left(I_2 + m_2r_2^2\right)\ddot{\theta}_2 - m_2r_2\sin(\theta_2)\ddot{x}_1 + m_2r_2\cos(\theta_2)\ddot{y}_1 + m_2r_2(l_1 - r_1)\cos(\theta_1 - \theta_2)\ddot{\theta}_1 \\
& - m_2r_2(l_1 - r_1)\sin(\theta_1 - \theta_2)\dot{\theta}_1^2 + m_2r_2g\sin(\theta_2) = \tau, \\[1.5em]
& (c) (m_1 + m_2 + m_3)\ddot{x}_G = \lambda_{n,2}^b, \\[1.5em]
& (d) (m_1 + m_2 + m_3)\ddot{y}_G = -(m_1 + m_2 + m_3)g + \lambda_{n,1}^b, \\[1.5em]
& (e) f_1(q) = y_1 - r_1 \sin(\theta_1) = 0, \quad f_2(q) = x_1 - r_1 \cos(\theta_1) = 0.
\end{aligned}
\right.
\qquad (C.4)
$$

These are equivalent (with admissible initial conditions) to (C.1). Similarly we can replace both $\ddot{x}_1$ and $\ddot{y}_1$ in (C.4) (a) (b) by their expressions obtained from (C.4) (c) (d), and we obtain transformed (C.4) (a) (b) as second-order dynamics of $\theta_1$ and $\theta_2$ with multipliers and $\tau$.

These developments show that (1) is an extension of systems with bilateral holonomic constraints, compare (C.2) and (C.4): the only thing which changes is the way the multipliers are calculated, and the constraints associated with them. In this particular case, it may be easier to use the reduced dynamics (C.1). However eliminating coordinates is not always trivial. If the goal is to control the center of mass, then (C.4) (c) (d) can be used to design $\lambda_{n,1}^b$ and $\lambda_{n,2}^b$ (both multipliers can be calculated [92] [6, section 5.1.1] as functions of the state and $\tau$), then inject this in the transformed (C.4) (a) (b) which provide the evolution of $(\theta_1, \theta_2, \dot{\theta}_1, \dot{\theta}_2)$ (this is similar to the developments in section 3.1.2, however here no sign constraints exist).

Let us examine conditions upon which the transformation (83) relies.

**Lemma 1.** Let $q_1(z_1, z_2)$ be a differentiable function. Assume that $\frac{\partial q_1}{\partial z_1}$ and $\frac{\partial q_1}{\partial z_2}$ are invertible. Then $M_{11}(z)\ddot{z}_1 + M_{12}(z)\ddot{z}_2 + F_{z_1}(z_1, z_2, \dot{z}_1, \dot{z}_2) = M_1(q_1)\ddot{q}_1 + F_1(q_1, \dot{q}_1)$ is fulfilled with $M_1(q_1) > 0$ only if $(\frac{\partial q_1}{\partial z_1})^{-1} M_{11}(z) = (\frac{\partial q_1}{\partial z_2})^{-1} M_{12}(z) > 0$.

**Proof:** assume that the equality holds. Then calculations yield $M_{11}(z) = M_1(q_1(z))\frac{\partial q_1}{\partial z_1}$ and $M_{12}(z) = M_1(q_1(z))\frac{\partial q_1}{\partial z_2}$. The result follows. $\square$

State space transformations that yield humanoids' centroidal dynamics are analyzed in [329, 330].

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