Human-Inspired Control of a Whip: Preparatory Movements Improve Hitting a Target

Abstract

Manipulating flexible and underactuated objects, such as a whip, remains a significant challenge in robotics. Remarkably, humans can skillfully manipulate such objects to achieve tasks, ranging from hitting distant targets to extinguishing a cigarette’s in someone’s mouth with the tip of a whip. This study explored this problem by constructing and modeling a 25-degree-of-freedom whip. Our goal was to investigate the strategies employed by humans when using a whip to strike a target. To that end, a human-inspired controller was devised that emulated two observed movement strategies: “striking only” and “preparing and striking”. While the latter strategy involved a more intricate and parameter-intensive trajectory definition, our findings revealed that the more complex “preparing and striking” approach enabled the whip to reach targets at greater distances. The outcomes of this study provided first insights into preparatory movements that humans employ when manipulating objects. By directly bridging between human and robot studies, we show how insights into human movements may inform effective robot control strategies for the manipulation of underactuated objects.

I. INTRODUCTION

Cutting-edge robot manipulation algorithms have revolutionized the way robots handle objects, granting them a remarkable level of dexterity and precision [1], [2]. Recent strides in end-effector design have further pushed the boundaries with the development of soft and adaptable grippers equipped with tactile sensors, expressly designed for the manipulation of objects of a wide variety of shapes [3], [4]. However, these successes applied to the manipulation of rigid objects. As robots embark on their journey into the unstructured human world, they will inevitably confront deformable objects, such as clothing, cables and ropes. This presents a significant challenge as traditional control algorithms, initially tailored for rigid object manipulation, fall short in effectively addressing the intricacies of non-rigid materials.

Two core obstacles arise when dealing with deformable objects: their inherently underactuated nature with high degree-of-freedom nonlinear dynamics. Flexible objects exhibit complex and unpredictable behaviors, which demand a fundamentally different approach for manipulation. Recognizing these challenges, recent research endeavors have aimed to overcome these obstacles with innovative strategies.

Recently, data-driven techniques, such as deep learning, have emerged as powerful tools to tackle the manipulation of non-rigid objects. Models based on machine learning can capture and adapt to the nuances of these objects, enabling robots to manipulate them with greater proficiency [5]–[14]. However, these approaches need huge amounts of training data and rely on physics engines to simulate the system requiring lengthy iteration procedures.

Another approach is the development of algorithms that emphasize slow, deliberate movements to account for the dynamic nature of flexible materials. These algorithms can be effective for stabilizing deformable materials, but may not fully harness their inherent dynamics. Deformable objects possess unique properties, such as compliance and configuration-dependent inertia, that can be leveraged to advantage in certain tasks. A truly intelligent manipulation algorithm would go beyond quasi-static approaches and dynamically exploit the passive dynamics of flexible materials to enhance task efficiency and accuracy.

While some tasks may benefit from quasi-static movements [15], [16], there are others that can only be achieved through high-speed dynamic movements [10], [11], [17]. For instance, understanding how a rope naturally coils or how clothing drapes could enable a robot to manipulate these materials with greater efficiency and finesse. Therefore, the next frontier in robot manipulation of deformable objects lies in the development of algorithms that can adapt in real-time to the dynamic behavior of such materials, ultimately fully exploiting their properties.

Compared to robots, humans are highly skilled in manipulation of flexible objects and able to perform highly dynamic movements [18], [19]. This study will take inspiration from human strategies to control underactuated objects, enabling dynamic movements that leverage the inherent passive dynamics of the object. To demonstrate our research approach, we chose a task that cannot be successfully completed through quasi-static manipulation - manipulating a whip to strike a target. Our research endeavors to create human-inspired control algorithms that enable robots to dynamically interact with a whip in a precise and dynamic fashion.

To develop our human-inspired controller, we examined human arm and whip kinematics during a target-hitting task. This revealed two prominent movement strategies: one with only a single hitting movement, and another that first prepared the whip with a counter-movement before striking the target, similar to a wind-up before a throw. We developed a controller that generated trajectories in hand space based on trajectory optimization. This work extends a previous simulation study by examining human performance and generating them with a modeled whip [20]. These trajectories were modeled in end-effector space shown to be primary for human control [21], [22]. With this control framework, we examined the benefits of preparing the whip prior to the targeted striking action. By modeling and controlling the whip with insights drawn from humans, we aim to bridge the gap between robotic approaches and human control, aiming to advance the field towards more effective and versatile manipulation of deformable objects in real-world scenarios.

II. METHOD

A. Participants and Data Collection

Five volunteers (23 - 30 years, 4 females and 1 male) were recruited to manipulate a 3D-printed whip to hit a target. They gave informed consent using protocol #16-02-05 approved by Northeastern University. Two targets were placed at 1.2m in front of the participant at two different vertical positions (.1 m above and −.2m below handle height in initial position (Fig. 2). At the start of each trial, the whip was stationary with all of its links hanging down toward the ground. To ensure the whip was fully extended without contacting the ground, participants stood on a raised platform. Participants performed hand movements along the x and z axes in the sagittal plane. Each participant carried out 20 separate trials for each target position. Reflective markers were placed on the participant’s shoulder, elbow, and wrist; each of the 25 whip links was marked by a small reflective patch of tape. Twelve Oqus 3+ cameras tracked the movements of the subject and the whip at a sampling frequency of 100 Hz (Qualisys, Goetheborg, Sweden).

Fig. 2.

Experimental setup used to collect participant data when manipulating the 3D-printed whip. Two different targets were placed in front of the participant; here showing the target at vertical height of −.2m.

B. 3D-Printed Whip

A 1-meter long whip was constructed from 25 distinct spherical links, each 3D-printed individually with a diameter 40 mm and mass approximately 7 grams. The links were connected by steel pins with a radius of 2mm and a length of 20 mm. The pins served as hinge joints in a serial arrangement, restricting the whip’s motion to a two-dimensional plane. To avoid friction between the pin and the link when the movements were not strictly planar, a small tolerance was introduced in the through-hole of the links (Fig. 1).

Fig. 1.

Fully assembled 3D-printed whip with 25 links. The enlarged sections highlight how the relative angles between the links were defined and where the reflective tape was placed.

C. Modeling the Whip Dynamics

To model the whip dynamics, we adopted an energy-based approach based on the Euler-Lagrange formalism with the configuration variable vector $q$ that embodied all joint angles $q_i$. Next, forward kinematics obtained the position $P_i$ and velocity $v_i$ of the point mass in each segment, as given by

$$P_n=P_{n-1}+R\left(\sum _{j=1}^{n-1}{q}_j\right)\left[\begin{array}{l}L\hfill \\ 0\hfill \end{array}\right]$$

$$v_i=v_{i-1}+\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]R\left(\sum _{j=1}^{i-1}{q}_j\right)\left[\begin{array}{c}L\\ 0\end{array}\right]\left(\sum _{j=1}^{i-1}{q}_j\right)$$

where $R$ is a 2D rotation matrix and $L$ denotes the length of each link, respectively. The total potential $V_{tot}$ and kinetic energies $K_{tot}$ were obtained based on $P_i$ and $v_i$:

$$V_{tot}=\sum _{i=1}^n{m}_{i}g{P}_i^v$$

$$K_{tot}=\sum _{i=1}^n\left(\frac{1}{2}{m}_i{v}_i^{⊺}{v}_i+\frac{1}{2}{\omega }_i^{⊺}I{\omega }_i\right)$$

where ${\omega }_i$ denotes angular velocity vector of i-th link; $m_i$ is the mass of i-th link; $g$ represents the gravity constant and is equal to approximately $9.81m/s^2$. The mass moment of inertia of i-th link is denoted by $I$ which is considered to be the same for all links. Based on the total kinetic and potential energies, the Lagrangian functional $ℒ$ was obtained by

$$\frac{d}{dt}\left(\frac{\partial ℒ}{\partial \dot{q}}\right)-\frac{\partial ℒ}{\partial q}=\Gamma$$

where $\Gamma$ denotes the generalized forces. From the above Lagrange formalism the mass-inertia matrix and the Coriolis terms can be obtained as follows. The mass-inertia matrix for the whip structure is given by

$$K_{tot}\left(q,\dot{q}\right)=\frac{1}{2}{\dot{q}}^{\top }M\left(q\right)\dot{q}$$

and the Coriolis term is given by resolving Christoffel symbols:

$$h\left(q,\dot{q}\right)\dot{q}=\left(\frac{\partial }{\partial q}\left(M\left(q\right)\dot{q}\right)\right)\dot{q}-\frac{1}{2}{\left(\frac{\partial }{\partial q}\left(M\left(q\right)\dot{q}\right)\right)}^{\top }\dot{q}$$

Using these terms the second-order differential equation in matrix form is obtained:

$$M\left(q\right)\ddot{q}+h\left(q,\dot{q}\right)=Bu+J^{⊺}\lambda$$

where $B$ is the input matrix obtained through the principle of virtual work, and $u$ is the control input vector used to simulate the stiffness and damping at each joint using a PD (Proportional-Derivate) controller. $J$ is the Jacobian matrix, and $\lambda$ is a vector of Lagrange multipliers. The Lagrange multipliers are recruited to constrain the handle movement as described in the next sections.

Although this approach could model whips with different stiffness, damping, and mass (for example, a tapered whip), this study started with a simplified structure. In the 3D-printed whip all links had the same mass, as well as minimal stiffness and damping. In the whip model, each link’s mass, stiffness, and damping were set to 7 grams, with stiffness $0.00001N\cdot m/rad$, damping $0.00001N\cdot m\cdot s/rad$ respectively. This simplification served to streamline the system identification process using prior knowledge of the model parameters.

D. Model Evaluation

The x- and z-trajectories of the whip handle obtained from participant data were used as the handle trajectory inputs to the Lagrangian model. Initial conditions for the whip’s links were set to zero, with the whip pointing downwards. MATLAB’s ode45, a non-stiff differential equation solver, was used to forward-simulate the whip model, obtaining the x- and z-trajectories for all 25 links.

The performance of the overall model was assessed by calculating the mean Euclidean error between the simulated whip’s trajectory and the actual whip’s trajectory. This evaluation employed the following formula:

$$Error=\sqrt{{\left(x_{sim}-x_{real}\right)}^2+{\left(z_{sim}-z_{real}\right)}^2}$$ (1)

In this equation, $x_{\mathrm{sim}}$ and $z_{\mathrm{sim}}$ represent the coordinates of the trajectory for each individual link of the simulated whip, while $x_{\text{real}}$ and $z_{\text{real}}$ are the corresponding coordinates of the real whip’s trajectory. Equation 1 was applied to all 25 links of the whip, resulting in an error average over time for each link.

E. Analysis of Human Movement Strategies

The kinematic data obtained from all 40 trials per participant were analyzed to determine the movement strategies used to strike the two targets. The arm movement was represented by the angle defined by the line between the handle and the shoulder (neglecting elbow flexion) and the downward vertical (0 deg).

F. Trajectory Generation

For human point-to-point movements it has been shown that velocity exhibits a bell-shaped profile that is best approximated by a trajectory that minimizes the third derivative of position (jerk), equivalent to maximizing smoothness [23]. Equation (2) generates such a trajectory from one point to another for a given duration.

$$\begin{gathered} x(t)=x_i+\left(x_f-x_i\right)\left(6{\tau }^5-15{\tau }^4+10{\tau }^3\right) \\ z(t)=z_i+\left(z_f-z_i\right)\left(6{\tau }^5-15{\tau }^4+10{\tau }^3\right) \end{gathered}$$ (2)

Here, $x$ and $z$ represent the coordinates of a linear 2D trajectory, where $x_i$ and $z_i$ are the initial positions, $x_f$ and $z_f$ are the final positions, and $\tau \left(\tau =t/t_f\right)$ denotes the normalized time duration of the trajectory; $t_f$ is the duration of the movement. Note that even though the movement is described using both the x- and z-coordinates, the motion itself remains linear within the xz-plane.

Based on Equation (2) and motivated by human data, two movement strategies can be generated:

1. Striking only: generate a single forward movement to strike the target.

2. Preparing and striking: combine a backward and a forward movement to strike the target.

The striking only strategy is a single point-to-point movement of the whip handle to strike the target with the tip of the whip. This strategy relies on five parameters: $x_i,z_i,x_f,z_f$, and $t_f$, to define the handle’s trajectory.

The preparing and striking strategy combines two minimum-jerk trajectories, i.e., first (preparing) and second (striking) movement denoted by subscripts 1 and 2 respectively. In the simplest case, the end position of the first movement ($x_{1,f},z_{1,f}$) becomes the start position of the second movement ($x_{2,i},z_{2,i}$). For more flexibility an additional parameter, overlap, allowed for partial overlap between the two movements. For example, with $25\mathrm{\%}$ overlap, the velocity of the last $25\mathrm{\%}$ of the first movement was superimposed on the velocity of the first $25\mathrm{\%}$ of the second movement. The whip handle’s position was determined by integrating the summed velocity trajectory. Thus, the preparing and striking strategy involved 9 parameters: $x_{1,i},z_{1,i},x_{1,f},z_{1,f},t_{1,f},x_{2,f},z_{2,f}$, and $t_{2,f}$, plus the overlap parameter.

G. Trajectory Optimization

An optimization algorithm was implemented to generate trajectories that hit the target with the modeled whip. MATLAB’s patternsearch algorithm was used to identify optimal values for the hand trajectory parameters that minimized the distance between the tip of the whip and the target. The cost function for the optimization was as follows:

$$cost=min\left({\left(x_{tip}-x_{\mathit{\text{target}}}\right)}^2+{\left(z_{tip}-z_{\mathit{\text{target}}}\right)}^2\right)$$ (3)

In this equation, $x_{\text{tip}},z_{\text{tip}},x_{\text{target}}$, and $z_{\text{target}}$ represented the respective coordinates of the whip’s tip and the target position throughout the striking attempt. Consequently, the optimizer’s objective was to ensure that, at some point during the movement, the whip’s tip accurately hit the target. Based on the human experiment, the bounds for the optimization parameters were −0.2m to 0.2 m for all x and z parameters, 1s to 2s for the duration parameters, and 0 to 100 for the percent overlap parameter. The patternsearch optimizer was configured with a maximum of 200 iterations, an initial mesh size of 0.0001, and a mesh tolerance of 0.0001.

H. Grid Search for Reachable Targets

To compare the effectiveness of the two movement strategies, a grid search was conducted to identify ”reachable” target positions for each strategy. This search covered an area measuring 0.9m by 1.2m, with a resolution of 0.05m extending to the right of the whip’s initial position. For each target position, this search space was swept iteratively to identify the optimal trajectory parameters for the minimum distance achieved between the tip of the whip and the target. This optimization was conducted for both the striking and preparing and striking strategies.

III. RESULTS

A. Human Kinematics and Two Strategies

Fig. 3A and B show representative trajectories of the handle and the tip of the whip in the sagittal plane from one participant. Note that the participants were manipulating the 3D-printed whip and with experimentally controlled initial position, hanging down and at rest. Panel A shows a single striking action, while panel B shows a back-and-forth movement, i.e., preparing and striking. Time progression is indicated by the arrows at the handle trajectories. Both strategies hit the target (shown as a black point). Panel C to F displays the corresponding time series of arm angle and velocity in the two strategies, with background shades highlighting the two phases in the second strategy. The striking action exhibits the bell-shaped velocity, or minimum-jerk profile typically seen in point-to-point movements [23].

Fig. 3.

Trajectories of the whip handle in two representative trials where one participant performed a single striking movement (A), and a back-and-forth movement (B). Handle trajectory (black line) and whip tip trajectory (red line) are shown for the full movement. The whip configuration is shown for 10 time samples during the trial (gray). Arm angle (C and E) and its velocity (D and F) are shown for the two movement strategies.

B. Model Validation

To validate the parameterized whip model, the handle trajectories from participants were used as input and the resulting trajectories of the tip and the 25 links of the modeled whip were compared with the tip and links of the real 3D-printed whip manipulated by humans. The model’s match was quantified by the mean error between the corresponding links of the modeled and the real whip (Eq. 1). Fig. 4 shows that the simulated whip’s kinematics closely matched those of the real whip in both x- and z-coordinates (Fig. 4A and B. The mean errors for each link are shown in Fig. 4C, exhibiting an average error of less than 5cm at the tip.

Fig. 4.

Comparison of the whip tip trajectory in x (A) and z (b) of the actual 3D printed whip (blue line) and the modeled whip (red line). C: Mean Euclidean error calculated for each link of the whip. Mean error for link 25 corresponds to the trajectories shown in panels A and B.

C. Comparison of Two Movement Strategies

To assess the utility of the back-and-forth movement, we developed a simplified model for generating handle trajectories in the sagittal plane. Using one or two minimum-jerk profiles with temporal overlap, the optimization procedure parameterized these trajectories to generate handle and whip trajectories that achieved the closest hit. Fig. 5 presents two sample trajectories from this optimization, generating a striking only and a preparing and striking action. The handle movements were confined to the area highlighted in gray, similar to how human hand movements were limited. For the shown target position, only the strategy that added the backward movement yielded a successful hit.

Fig. 5.

Kinematic data of the whip handle in two trials obtained through optimization with a striking only strategy (A), and a preparing and striking strategy (B). Handle trajectory (black line) and whip tip trajectory (red line) are shown for the full movement. The whip configuration is shown for 10 time samples during the trial (light blue).

To better understand the benefits and shortcomings of the two control strategies, a grid search was conducted to determine which target locations were reachable with each approach. Fig. 6A and B shows the minimum distance between the tip and the target achieved for the grid of target locations, highlighting the regions of the targets that could be reached (in blue) and the targets that could not be reached (in red). Fig. 6A shows the results for the striking only strategy, Fig. 6B the preparing and striking strategy, Fig. 6C overlaps the locations showing hits in the striking only strategy in light green and the preparing and striking strategy in dark green. A hit was defined as the whip tip being within 1cm of the target. The backward movement significantly increased the reach of the whip and allowed to access farther targets compared to the single striking only movement strategy.

Fig. 6.

Minimum distance between whip tip and target achieved for target positions using striking only (A) and preparing and striking (B) strategies. Color indicates the error. C: Binary hit map for the two strategies. An error of less than 1cm between the whip tip and target was considered a hit. Farther targets were only hit with the preparing and striking strategy.

IV. DISCUSSION

This study examined how humans manipulate a whip to strike a target with the goal to develop a robot controller that similarly controls a flexible object in a target-oriented fashion. To facilitate the match between humans and robot, a simplified whip with 25 links was 3D-printed and modeled using the Lagrangian formalism. A main observation in human performance was that humans tended to employ strategies with a backward movement prior to the striking action. This observation was mimicked in a control model based on simplified handle movements, i.e., primitives [24]. Results showed that the seemingly superfluous backward movement increased the range of target distances that were reachable.

A. Significance of Preparatory Movements

In many daily activities and sports, humans perform preparatory movements before executing the main task. An illustrative example is seen in golf, where players execute a backswing before the forward swing to hit the ball. Research has shown that such preparatory actions “preload” or stretch the trunk muscles, enabling golfers to generate more power and strike the ball more forcefully [25]. A similar observation holds for baseball pitching, where the “wind-up” phase aligns body movements, contributing to a more powerful energy release during the throw [26]. This principle extends to other sports like handball, where the wind-up influences ball velocity during the throw [27]. In addition to the biomechanical benefits of preparatory movements, muscle-physiological arguments have been made that stretching the muscles prior to the focal action enhances control [28].

The backward movements in the whip task, however, may hold one more benefit: the underactuated object is prepared both dynamically and geometrically to facilitate its subsequent manipulation. Our model simulations underscored one key advantage in that farther targets could be reached. This is likely due to the additional energy imparted to the whip system before striking. This ‘stored’ energy, combined with the energy added during the striking phase, enhanced the whip’s reach. Further, the object may be arranged into favorable initial conditions that may align the body and whip, facilitating more consistent and precise target hitting. The importance of ”arranging” the whip before the forward striking actions was shown in human data on a previous study [18]. Trials where the whip was extended and at a certain azimuth angle before the strike were correlated with smaller hitting errors. Proper whip arrangement might also lessen sensitivity to variability and external disturbances, although further research is needed to confirm this.

B. Limitations

The 3D-printed and modeled whip designed for this study featured uniform mass distribution with very small values of stiffness and damping, unlike real whips. However, this simplification provided a stepping stone for investigating more intricate whips and whip-like structures, incorporating variations in mass, stiffness, and damping across different segments. A variety of algorithms available for system identification in both linear and nonlinear systems can be used in future work [29]. By comparing modeled to experimental performance with a real whip, these methods can adjust model parameters to closely match actual whip behavior.

V. CONCLUSION

This study demonstrated how human experiments can provide a foundation for exploring more insights into control strategies to be implemented on a robot. This work exemplified the sequence of creating an object that permitted an exact modeling to enable matching between human and simulated performance that can be implemented on a robot. The specific observation of object preparation has found little consideration in robotics to date and may be a contribution to enable robots to dynamically handle complex objects.