Human control of underactuated objects: Adaptation to uncertain nonlinear dynamics ensures stability

Abstract

Humans frequently interact with objects that have dynamic complexity, like a cup of coffee. Such systems are nonlinear and underactuated, potentially creating unstable dynamics. Instabilities generate complex interaction forces that render the system unpredictable. And yet, humans interact with these objects with ease. Nonlinear dynamic analysis shows that the initial conditions and frequencies of input forces determine the system’s stability. Taking inspiration from carrying a cup of coffee, participants rhythmically moved a cup with a ball rolling inside which was modelled as a cart-pendulum system. They were encouraged to prepare the cup-and-ball system by ‘jiggling’ the cup before moving it back and forth on a horizontal line. We tested the hypothesis that humans initialize the system and choose interaction frequencies that stabilize their interactions. To create uncertainty about the specific cup-and-ball system, the pendulum length was varied without providing cues to the participant. Stability was quantified by variability of relative phase between cup and ball. Results showed that participants nonlinearly co-varied the initial ball angle at the end of preparation and the cup frequency during the rhythmic phase. Mapping participants’ choices onto the highly nonlinear manifold of stable solutions generated by forward-simulations verified that they indeed achieved stable solutions.

I. INTRODUCTION

Carrying a plate of food or a cup filled with coffee are simple tasks for humans. Yet, despite the apparent ease with which humans complete such actions, modern-day robots are still unable to replicate this performance. Considering the task of transporting a cup of coffee, humans and robots face significant dynamical challenges. The fluid dynamics of the coffee can exhibit chaotic and unpredictable behavior due to the nonlinearities. The lack of direct control (i.e., underactuation) over the coffee’s dynamics can result in undesirable forces acting on the cup and on the hand. Previous work showed that humans account for unpredictable system dynamics and uncertainty in system characteristics [1]. However, aspects like the mass of the cup or the amount of coffee in the cup can also be uncertain when picking up a new cup. And yet, humans easily adapt to new scenarios. Understanding the motor control principles that underpin such mostly successful human interactions can inform the development of control policies for robots.

Research in human motor neuroscience has largely focused on control strategies in unconstrained free movements or when reaching or grasping rigid objects. Only a handful of studies examined human interactions with complex systems, like a mass-spring or a cart-pendulum system, that have internal degrees of freedom. Several studies used a linear mass-spring system to assess the roles of visual and/or haptic feedback for state estimation and dynamic feedback-based control [2], [3]. Intermittent, continuous, predictive control, or perceptual control models have been posited to capture human motor control when balancing an inverted pendulum mounted on a cart [4]–[6]. Several studies investigated optimal control theory to model human control of underactuated systems by considering optimization criteria such as smoothness, effort and/or accuracy [7], [8]. The above mentioned control approaches rely on internal models that are difficult to estimate when systems are highly nonlinear, underactuated or deformable, and have unidentified system characteristics [9]. Moreover, predictive approaches that are required to compensate for sensory noise and delays work poorly when the dynamics turn chaotic and unpredictable.

Hence, given these challenges, this study aims to identify the factors that determine the efficacy of interactions with nonlinear underactuated systems. One fundamental goal for robotic controllers is to maintain dynamic stability. Rhythmic interactions in particular have the potential to cause asymptotically growing instabilities and unbounded interaction forces; the controller needs to prevent amplifying chaotic and unpredictable behavior. For nonlinear systems, both amplitude and frequency content of the input signal can influence system’s response characteristics [10]. Furthermore, initial conditions of the system can significantly affect the time it takes to reach steady behavior or, in the worst case, lead to chaotic patterns [11]. Thus, humans need to tune their input frequencies and initial conditions to ensure predictable and stable dynamics when interacting with nonlinear and underactuated systems.

Past work of our group investigated human interactions with a dynamically complex object inspired by carrying a cup of coffee. The dynamics of sloshing coffee was simplified to the model of a ball (a point mass) moving inside a semi-circular cup. It was shown that humans prepare initial conditions of the cup-ball system before executing rhythmic back-and-forth movements of the cup to ensure predictable and stable dynamics of the ball [12], [13]. Experimental results on predictability and stability were confirmed by conducting forward simulations of the cup-ball system using a simple impedance controller. However, the participants were constrained to use a particular interaction frequency and the system characteristics were constant. Thus participants were able to converge to initial conditions that ensured system predictability and stability.

The present work utilized the cup-ball paradigm to examine how participants adapted when they were faced with cup-ball systems of different unknown dynamics. Participants were encouraged to prepare the cup-ball system before executing rhythmic back-and-forth cup movements on a horizontal line. We tested the hypothesis that humans flexibly prepare the system and choose a frequency that stabilizes their interactions with the nonlinear underactuated system. Comparing the experimental results with forward simulations of the cup-ball system using an inverse dynamics controller revealed that subjects varied their input in nonlinear ways to achieve dynamic stability.

II. METHODS

A. Participants

Ten college students (19 to 35 years, 4 females) participated in the experiment. All were right-hand dominant and had no history of neurological or biomechanical issues. Experimental procedures were explained to each participant prior to the study. Participants gave written informed consent as approved by Northeastern University’s Institutional Review Board (IRB#:10-06-19).

B. Dynamical Model

Participants were tasked with transporting a “cup of coffee”, simplified to a ball sliding inside a circular arc (Fig. 1A). This cup was constrained to move on a horizontal line delimited by two target boxes. The cup-ball system was mechanically equivalent to a cart with a suspended pendulum where the cart and the pendulum bob represented the ball and cup, respectively. The angular position of the ball was equivalent to the angle of the pendulum. Despite this simplification, the essential features of underactuation and nonlinearity in the dynamics of “carrying a cup of coffee” were retained. The following dynamical equations describe the cup-ball system:

$$\left(m_c+m_b\right)\ddot{x}=m_{b}l\left({\dot{\theta }}^2\mathit{\text{sin}}\theta -\ddot{\theta }\mathit{\text{cos}}\theta \right)+F_{\mathit{\text{applied}}}=F_{\text{ball}}+F_{\mathit{\text{applied}}}$$ (1)

$$\ddot{\theta }=-\frac{\ddot{x}}{l}\mathit{\text{cos}}\theta -\frac{g}{l}\mathit{\text{sin}}\theta$$ (2)

where $x$ is the cup position and $\theta$ is the ball angle. The ball position at the bottom of the cup (downward vertical orientation of the pendulum bob) is defined as zero deg, with counterclockwise direction defined as positive. The mass of the cup is $m_c=2.8\text{kg}$, the mass of the ball is $m_b=0.6$ kg, and the gravitational acceleration is $g=9.81m/s^2$. The length of the pendulum $l$ was set to three different values: 0.3 m or 0.6 m or 1.2 m. $F_{\mathit{\text{applied}}}$ is the force applied by the participant’s hand on the cup. $F_{\mathit{\text{ball}}}$ is the force applied by the ball on the cup.

Fig. 1: Experimental Task and Protocol.

A Coffee sloshing inside a cup was simplified to a ball rolling inside a cup. If the ball slides, instead of rolling, the system is mechanically equivalent to a cart-pendulum system. B Participants controlled the cup-ball system shown on a screen by controlling the handle of a robotic manipulandum. C The task was to move the cup horizontally between the two green blocks in a rhythmic fashion. The cup position $x$ was measured relative to the midpoint between the two green blocks (rightward positive); the ball angle $\theta$ was measured with respect to downward vertical (counterclockwise positive). D Cup position and ball angle of a single participant in the preparation and rhythmic phases. E Over two experimental sessions, participants completed 5 blocks of 42 trials each (14 trials × 3 pendulums) with pseudorandom pendulum order within the block.

C. Experimental Task and Procedures

The task was realized in a virtual environment and displayed on a large projection screen facing the participant. The cup was displayed as a blue circular arc and the ball as a red ball (Fig. 1B,C). Regardless of the pendulum length, the visual scale of the cup and ball remained the same (22.5 cm radius) to conceal the specific system to the participant. A thin grey horizontal line representing the floor was bounded by two rectangular green target boxes, Box A and Box B, their centers at −0.15 and +0.15 m. Participants sat on a chair in front of the screen and gripped the handle of an admittance-controlled robotic manipulandum (HapticMaster, Motekforce, Netherlands, Fig. 1B). The time difference between the moment the robotic handle is moved until the movement is displayed on the screen is 33 ms ±17 ms, which is a typical visuomotor delay for robotic interfaces. Participants controlled the position of the cup, $x$, by moving the robotic handle laterally. The position of the chair was adjusted such that the participant could comfortably grasp the handle and move the cup between the two rectangular boxes. The motion of the robotic handle was limited to horizontal translations. The ball force $F_{\mathit{\text{ball}}}$ was fed back to the participant’s hand via the robotic handle. The applied force on the cup $F_{\mathit{\text{applied}}}$, the position $x$, velocity $\dot{x}$ and acceleration $\ddot{x}$ of the cup, and the computed angular position $\theta$, velocity $\dot{\theta }$, and acceleration $\ddot{\theta }$ of the ball were recorded at 120 Hz using a custom written C++ program.

At the beginning of each trial, the cup was in Box A with the ball at rest at $\theta =0$ deg. Each trial had two phases: i) the preparation phase and ii) the rhythmic phase. In the preparation phase, participants were encouraged to explore the cup and ball dynamics by ‘jiggling’ the cup between the left block (−0.15 m) and the midpoint at 0 m. The exploration time was not restricted. When participants were ready to start the rhythmic phase, they moved the cup towards Box B. A bell sound was played when participants reached Box B for the first time. In the rhythmic phase, participants oscillated the cup between the two target boxes at their choice of frequency. After 15 s a bell sound indicated the end of the trial. An example trial showing cup position, ball angles across the entire trial, and the preparation and rhythmic phases is displayed in Fig. 1D.

D. Experimental Protocol

Participants completed the experiment over two experimental sessions on consecutive days, each lasting approximately one hour (Fig. 1E). At the beginning of the first session participants completed 10 familiarization trials with each of the three pendulum lengths ($l=0.3,$, 0.6 or 1.2 m) in a blocked fashion. The block order was counterbalanced between participants. Participants then performed two blocks of 42 trials (14 trials × 3 pendulums), each with pendulum lengths presented in random order. No external cues were provided that explicitly revealed the pendulum length on a trial. In the second session on the next day, participants completed 3 blocks of 42 trials each with pendulum lengths presented in random order. Across the two sessions, there were 80 trials for each pendulum length. The order of presented pendulum lengths was unique for each participant.

E. Data Analysis

For each trial, the time at which the cup velocity was zero before moving the cup into Box B was identified. This time point, defined as zero time, represented the start of the rhythmic phase. The portion of the trial before this time point was the preparation phase. The ball angle at this time point defined the initial ball angle resulting from the exploratory actions during the preparation phase. NOTE: This study only reports results from the rhythmic phase.

1). Rhythmic Interaction - Cup Frequency:

A primary variable of interest was the chosen cup frequency. It was calculated as instantaneous frequency, because this point approach allowed for deviations from a constant frequency. Using the Hilbert Transform, the mean-adjusted cup position data was first converted into instantaneous phase that was then numerically differentiated to render instantaneous frequency [14]. Cup frequency was quantified as the mean of this instantaneous frequency signal.

2). Dynamic Stability - Relative Phase Variability:

Relative phase between the cup position and ball angle characterized how the participants controlled the cup and ball relative to one another within a trial. To obtain relative phase, the instantaneous (Hilbert) phases of cup position and ball angle were unwrapped and subtracted from each other. A relative phase of 0 or 180 deg indicated in-phase or anti-phase relation between the cup and ball, respectively. Variability of the continuous relative phase signal was used as a proxy to quantify the stability of relation between the cup and ball kinematics. This metric was previously used in work on coupled oscillations [15]. Since the relative phase was distributed around the circle, the circular variance of the relative phase was computed and denoted as relative phase variability [16]. Relative phase variability ranged between 0 and 1, and smaller values (less spread) indicated greater stability.

F. Dynamical System Analysis using Model Simulations

To gain insight into the control of the simplified cup-ball system, we used an inverse dynamic controller to model a participant controlling the cup [17]. The controller was designed to oscillate the cup according to the desired cup position $x_d=-\mathit{\text{ACos}}\left(2\pi \mathit{\text{ft}}\right)$ [12], where $A=0.3m$, $f$ is the oscillation frequency of the cup and $t$ is simulation time. The initial velocity of the cup was set to zero. The initial ball angular velocity was set to $-4\mathit{\text{deg}}/s$ corresponding to the mean initial ball velocity at the start of the rhythmic phase across all participants. The dynamical equations of the cup-ball system were transformed to the following general form:

$$M(q,\dot{q})\ddot{q}+G(q,\dot{q})=F$$ (3)

where $q=[x,\theta ]$ is the vector of generalized coordinates of the cart-pendulum system, $M$ is the inertia matrix, $G$ is the vector consisting of gravity and Coriolis terms, $F$ is the control input to the system. The inverse dynamics controller then has the following form:

$$\begin{gathered} F=M(q,\dot{q})a+G(q,\dot{q}) \\ a=\ddot{q}+K_{P}e+K_D\dot{e} \end{gathered}$$ (4)

where $\dot{q}$ is the vector of desired accelerations $[\ddot{x},\ddot{\theta }]$, $e$ is the vector of tracking errors of the states $q-q_d=[x-\left.x_d,\theta -{\theta }_d\right],\dot{e}$ is the vector of tracking errors of the first derivative of the states $\dot{q}-\dot{q_d}=\left[\dot{x}-\dot{x_d},\dot{\theta }-\dot{\theta }\right],K_P=70$ and $K_D=40$ are PD (Proportional-Derivative) controller gains. Here, we set the desired states of the ball such that the portion of the controller input acting on the dynamics of the ball (second value in the control vector) was zero.

The rhythmic system behavior was simulated with different initial ball angles ${\theta }_0$ (−90° to 90°) and cup oscillation frequencies $f$ (0.3 Hz to 1 Hz). The total simulation time was 15 s in accordance with the experiment, and the step size was 10 ms. A 4th-order Runge-Kutta integrator was used to forward-simulate the system starting from the initial conditions by applying the control inputs $F$. We computed the relative phase metrics as described before.

G. Statistical Analysis

A first focus was the effect of different pendulum lengths (short, medium and long) on cup frequency and relative phase distributions. Kolmogorov-Smirnov tests were used to detect pairwise differences between pendulum conditions. To analyze the effect of pendulum conditions and practice on relative phase variability, a two-way ANOVA was conducted with pendulum length (short, medium, and long) and trials (1 to 80) as fixed factors. For posthoc analysis of the interaction effects, a pairwise trend analysis was conducted with pendulum lengths as a factor and trials as a covariate. The aov and emtrends functions in $R$ were used to run the ANOVA and the trend analysis, respectively. We used a significance threshold of $\alpha =0.05$.

III. RESULTS

A. Relative Phase

A representative trial from one participant interacting with a short pendulum ($l=0.3m$) is shown in Fig. 2A. The cup position shows very regular rhythmic movements between the two targets, as instructed; the ball angle similarly oscillates, although with slightly more irregular amplitudes. The relative phase between the cup and the ball was close to zero deg, i.e., an in-phase relation.

Fig. 2: Relative Phase.

A Cup positions (first row) and ball angles (second row) from the rhythmic phase of one short pendulum trial plotted against time, using the left y-axes. The corresponding Hilbert phases of the cup position and ball angle are plotted against time using the right y-axes. Unwrapped cup and ball phases (third row, left y-axis) were subtracted from each other and transformed between −90 deg and 270 deg to obtain the relative phase (third row, right y-axis). The cup and ball trajectories are mostly in-phase as indicated by the relative phase being close to zero. The force applied by the participant on the cup is plotted against time (fourth row, left y-axis). B For each pendulum condition, a probability heatmap of relative phase from all time samples across all participants is plotted against trials in gray scale in the background. Mean relative phases for all participants (different colored lines) are plotted against trials in the foreground.

The relative phase between cup and ball for all three pendulums and participants is shown in Fig. 2B. The mean relative phase from each participant (different colored lines) is shown as a function of trial number in the foreground. Several participants flexibly used in-phase or anti-phase strategies between the trials with the long pendulum. A probability heatmap of relative phase from all time samples across all participants is shown in grayscale as background. The summary histograms on the right vertical axis highlight that relative phase was mostly in-phase (concentrated around 0°) for the short and medium pendulums, while for the long pendulum both in-phase (0°) and anti-phase (180°) behavior was observed with similar frequencies. Kolmogorov-Smirnov tests indicated significant differences for all pairs of the three pendulum conditions ($D>0.21,p<0.001$).

B. Stability: Variability of Relative Phase

The variability of relative phase for the different pendulum conditions are shown across trials in Fig. 3A. The average values across trials for all participants is shown in Fig. 3B. Relative phase variability appeared lowest for the short pendulum in comparison to the medium and long conditions, although the statistical comparison via a two-way ANOVA did not reach significance. Only the interaction between the pendulum lengths and trials $\left(F\right(\mathrm{2,2367})=7.20,p<0.001)$ was significant, together with a main effect of trials. This pattern underscored a decline in variability with practice $\left(F\right(\mathrm{1,2367})=107.18,p<0.001)$. Pairwise comparison showed that the long pendulum displayed a significantly more negative trend than the short pendulum condition $\left(t\right(2367)=-3.78,p<0.001)$.

Fig. 3: Relative Phase Variability.

A Relative phase variability plotted as a function of trials for the three pendulum conditions. The familiarization trials (first 10 trials) are indicated by a grey box drawn in the background. Error ribbons indicate ± 1 standard error between participants. B Mean relative phase variability across trials for each participant (open circles). Box plots show 25^th, 50^th, and 75^th percentiles. Gray lines connect data within a participant.

C. Initial Ball Angles and Cup Frequency

Initial ball angles across trials are shown for the different pendulums in Fig. 4A. There was no noteworthy trend or signs of convergence across participants (in different colors). Fig. 4B shows the data for cup frequency across trials for all participants. A probability heatmap of cup frequency from all time samples across all participants is shown in gray scale in the background. The mean cup frequency across time samples within the trial are plotted for each participant as a function of trial in the foreground. The summary histograms on the right vertical axes indicate that the distributions were mostly unimodal for the short and medium pendulums, and bimodal for the long pendulum. Kolmogorov-Smirnov tests indicated significant differences for all pairs of the three pendulum conditions ($D>0.16,p<0.001$). However, no other trends were observed.

Fig. 4: Preparation and Rhythmic Behavior.

A For each pendulum condition, initial ball angle (at the start of rhythmic phase) plotted as a function of trials for all participants (shown in different colors). B For each pendulum condition, a probability heatmap of instantaneous cup frequency samples from all participants is plotted across trials in gray scale in the background. Mean cup frequency of a trial is plotted for all participants (different colored lines) in the foreground.

D. Interplay Between Initial Ball Angle and Cup Frequency

These variables did not display visible systematic patterns, and yet, participants established relatively steady, i.e. stable behavior, as exemplified in Fig. 2A. Therefore, a final analysis examined how the main variables related to relative phase variability. Fig. 5 displays the two primary variables, initial ball angle and cup frequency, mapped into relative phase variability for different pendulum conditions. Importantly, initial ball angle and cup frequency was sufficient to generate forward simulations that rendered computed estimates of relative phase. The heatmap of relative phase variability for all combinations of initial ball angles and cup frequencies computed for each pendulum length is shown in gray scale in the background. The mean cup frequency and mean initial ball angle from all trials for each participant (different colored dots) are overlaid on the heatmap. The figure shows an intricate nonlinear relation between these variables. Interestingly, participants selected initial ball angle and cup frequency such that the relative phase variability was close to the areas of minimum variability. Further, this selection was irrespective of the relative phase. In summary, there was an interplay between preparatory and steady state strategies to maximize the stability of cup-ball dynamics.

Fig. 5: Interplay of Preparatory and Rhythmic Behavior for Stability.

For each pendulum condition, a heatmap of relative phase variability obtained from forward simulations of the cart-pendulum system for combinations of initial ball angle and cup frequency is plotted in gray scale in the background. Initial ball angle and mean cup frequency from all trials (dots) and participants (different colors) is overlaid.

IV. DISCUSSION

This study investigated human control of underactuated nonlinear objects whose characteristics are not known a priori. Participants moved the cup-ball, i.e., cart-pendulum system with different pendulum lengths, after having explored its characteristics and prepared for the upcoming rhythmic task. Participants adopted initial ball angles and cup frequencies in a seemingly unsystematic fashion. Nevertheless, they established a stable relation between cup and ball that was in-phase with the short and medium pendulums, and both in- and anti-phase with the long pendulum. Variability of relative phase, an experimental proxy for dynamic stability of the cup-ball oscillations, showed similar degrees of stability for all three pendulums. Simulations with a simple inverse dynamics controller showed that the initial ball angles and interaction frequencies co-varied in a nonlinear fashion that ensured stability of the cup and ball dynamics. These findings revealed that humans are exquisitely sensitive to stability properties of the underactuated systems, even when facing different dynamic characteristics.

Attaining and maintaining stable interactions with underactuated nonlinear systems is challenging because of their potential for chaotic and unpredictable behavior. Safe interactions with a complex object, such as a cup of coffee, may be ascribed to control based on an internal model of the object. Interaction forces could be predicted and preempted, or, with a very fast feedback loop, compensated for in time to prevent “spillage” of coffee [4], [7]. Previous work showed that predictability of the object dynamics increased with practice [12]. Such predictability may also be achieved by establishing stability, because stable dynamics implies that any small perturbations return to the attractor, i.e., die out by itself [18]. Following previous work on predictability, this study focused on dynamic stability estimated and approximated by the variability of relative phase.

Past work on the cup-ball task has shown that humans select arm impedance and exploit the resonant frequencies of the cart-pendulum-arm system [19], [20]. The present study further demonstrated that humans flexibly adapted their interaction frequencies to different response characteristics associated with the three pendulum lengths. A preparation phase also allowed participants to adjust the initial conditions of the cup-ball system that previously showed to facilitate rhythmic interactions [12]. However, each behavioral feature, i.e., preparation or rhythmic interaction, on its own appeared much less systematic than in previous studies that only scrutinized interactions with a single object. Due to the random presentation of three different objects in this study, such convergence to optimal solutions seemed elusive at first. However, when examining their interrelation with respect to the degree of stability of the system, simulations with a simple controller created a solution space that revealed interesting nonlinear patterns. Human participants subtly adjusted behavioral variables to align with the manifolds that guaranteed minimal variability of relative phase. With longer periods of interaction, participants could converge to specific regions on the high-stability manifold that offers greater tolerance for system preparation and interaction frequencies.

Such a flexibility in motor strategies is consistent with findings from the human literature on motor variability showing that the human motor system preferably controls variability in dimensions that affects task performance [21], [22]. Reinforcement learning [23] and optimal control [24] frameworks have mathematically modeled this apparent flexibility in motor learning and control. Note, the forward simulations used here employed an extremely simple feedback controller to create the solution spaces for the rhythmic interaction. A more advanced controller might be needed to account for exploratory and preparatory strategies. Yet, we put primacy on simple control without the need for an internal model of the object that might be challenging for humans.

Stability of a dynamical system cannot be attained from the very start of the motion. Hence, the start-up phase plays an important role in establishing stability of a complex systems [12]. Our previous research has revealed that preparation indeed led to initial conditions that favored predictability. Even in continuous rhythmic manipulation of a highly complex object, a bull-whip, we showed that each strike was preceded by extending and aligning the whip to facilitate accurate striking of a target [25]. Edraki and colleagues further explored the benefits of preparing the whip dynamics before striking a target using a simulation approach [26]. In that study, preparatory counter-movements injected energy into the object to reach more distant targets. Note also, exploration and preparation should not be confounded. Exploration identifies the solution landscape, whereas preparing the object brings the system to favorable initial conditions. Moreover, initial conditions can be selected flexibly with preparation for the case when the system parameters are uncertain. It would be interesting to investigate the interactions during preparation and their effect on the choice of initial condition and interaction frequencies in the rhythmic phase. Hence, future work will more closely scrutinize how participants explored and prepared the cup-ball system before entering the rhythmic phase.

To understand human strategies in this complex task, we adopted a task-dynamic approach that focused on first understanding the task’s physical determinants and its affordances for control, without any assumptions about the controller [1]. The virtual rendering of the object and haptically-mediated interactions aided in minimizing environmental influences to fully characterize the system as it responded to the human control input. By simplifying the task to the well-known cart-pendulum system, different input-output relations could be tested with respect to the instructed rhythmic cup movements. This methodological approach has been adopted in a variety of interactive tasks, from throwing to bouncing a ball, allowing quantitative manipulations and perturbations to gain insight into human control [20], [27], [28]. As this approach uses behavioral data to characterize the object dynamics and the control input, these human findings allow transfer to control of robotic systems.

V. CONCLUSIONS

Our results highlight that system properties can be highly nonlinear, but that humans are strikingly sensitive to these and exploit them in their interactions to attain stability. Dynamic stability in the interactions appears as a pervasive objective, corroborating previous findings in other interactive tasks. Note that these analyses first analyze the task and its affordances and remain agnostic about the controller. Only after formulating the underlying dynamics of the object and the task, different controllers can be tested. Given the behavioral level of our task-dynamic analysis, the methods are applicable to robotics and may help advancing robotic control of underactuated objects. To our knowledge, specifically the aspect of object preparation has found little consideration in robot control of complex objects to date.