I Introduction
The dynamic dexterity of animals is far beyond the current capabilities of artificial systems, despite their theoretical limitation in information transmission and processing [1, 2, 3, 4]. An important limitation in stateoftheart control frameworks is the reduced performances and reliability when dealing with highly variable dynamic interaction (e.g., contacts). Here, a major limitation is the reliability of their stability upon having an accurate model of the interaction dynamics that is not easy to track when dealing with the intrinsic unpredictability of realworld scenarios [5]. Therefore, the identification of a method to overcome such limitation may have a positive impact on the development of legged robots such as quadrupeds and humanoids, haptic technologies, exoskeletons, and rehabilitation systems.
One of the theories on how nature has overcome these biological limitations is based on the theory that the body motor control acts as a hierarchical architecture of semiautonomous controllers [6, 4]. In this type of architecture, each layer is accountable for the stability while accurately executing the behaviour directed from the higher level controller, while at the same time coordinating and verifying the behaviour of lower level controllers. This can be observed starting from the muscular level where each muscle contains structures that can modulate its mechanical impedance [7, 8, 9, 10]. Moving to a higher level, motor synergies can be identified that coactivate muscles, often across multiple joints, to produce desired body motions [4, 2, 1, 6, 11]. The motor synergies themselves are identified at different hierarchies starting from a single agonist/antagonist behaviour to more complex stereotyped movements that are implemented in the coordination of complex actions, such as balance [4]. In summary, there is a multitude of studies indicating that the motor control modulates the mechanical impedance of the human body to adapt to the task requirements.
The identification of the motor control ability to modulate the body impedance lead to theories based on the PortHamiltonian framework [12]. However, their integration in a complex hierarchical architecture of semiautonomous controllers is still an open issue. Furthermore, despite interaction control based on the PortHamiltonian approach has been proposed in the forms of impedance and admittance control, there are still issues when dealing with variable impedance and the interaction with complex external dynamics.
Impedance and admittance control differ on the methodology used to decode the physical information exchanged from the system with the environment [12]. Impedance control transforms an incoming flow (i.e., velocity) into the desired effort (i.e., force), while admittance control transforms the incoming effort into the desired kinematics [12]. The stability of PortHamiltonian controllers relies on the ability to track their energy exchange, which is challenging when dealing with redundant systems and changing environmental conditions. The approaches for tracking the energy are based on projectors [5]. This makes these controllers susceptible to changes in the task (e.g., contact, admittance, and impedance changes) as well as singularities which is where these projectors can degenerate numerically.
The Fractal Attractor has been recently proposed to solve this issue by using a passive variable impedance. The controller stability is guaranteed by its passivity, which is maintained using the spring energy map of the system state [13, 14]. The damping component of the controller acts only as an energy sink (i.e., reference velocity set to zero). The controller relies upon an anisotropic variable impedance redistributing the energy absorbed during divergence to converge to the desired state rather than a limit cycle. Other benefits of this controller are that (a) the impedance is defined based on a desired force/displacement behaviour, (b) it can be safely tuned online, (c) multiple controllers can be superimposed without affecting stability, and (d) it can be calibrated to account for the physical limitation of the robot guaranteeing the controller global stability. The current formulation has two major limitations which are (1) the lack of a feedback loop on the force, and (2) the lack of force homogeneity when the controllers switch between divergence and convergence phases.
This manuscript proposes a new impedance profile formulation inspired by the agonist/antagonist configuration of the muscles to solve the force homogeneity issue. We introduce a forcefeedback, also occurring in biological actuation, implemented using a reinterpretation of the parallel force/position control force feedback in traditional impedance controllers. The controller is validated in a Simulink (Mathworks Inc, USA) system simulation to evaluate its ability to adapt its behaviour to the interaction with unknown environmental dynamics.
Ii Fractal Controller
The Fractal Impedance Controller is based on an attractor that alters its topology between its divergence and convergence phases [13]. During divergence, the conservative component of the impedance is described as a spring centred in the desired position, while during convergence it is substituted by a nonlinear spring that has a stiffness profile equivalent to a constant spring centred in the midpoint between the desired position and the maximum displacement () reached during the divergence phase, producing the attractor in Fig. 2 [13].
This approach has been inspired by the agonist/antagonist muscles and the concept of dynamic primitives, which can be seen as libraries of stereotyped basic behaviours that can be superimposed to achieve complex actions [1, 2, 15, 4, 16]. Specifically, the hypothesis was that it should be possible to embed stability into the control algorithm itself identifying a stereotyped behaviour that could scale with the energy accumulated in the system without compromising stability. In other words, is it possible that what we observe as variable impedance in humans is produced by the scaling of intrinsically stable stereotyped embedded systems?
Following such a change of perspective raised the problem of how to embed a smooth trajectory into the attractor of the impedance controller itself. The implemented solution came from nature and relies on the trajectory of the limit cycle of a constant spring set halfway between the desired state and the maximum displacement reached during divergence [13]. This solution enabled the development of an impedance controller that can be tuned setting the desired force/displacement characteristics. However, these characteristics currently come at the cost of a sudden change in force magnitude while starting to converge and the lack of force feedback loop required for tasks, such as haptic exploration.
Iia Altering Fractal Impedance Profile for Force Homogeneity
The fractal attractor proposed in [13, 14] uses the conservation of energy principle to calculate the desired impedance for convergence, leading to a change of the gradient of energy at the inversion point that results in a sudden change in the exerted force. A solution to this issue may be provided by considering an agonist/antagonist control strategy, in which the two systems are impedance connected in parallel and the total produced effort (i.e., force) can be corrected by modulating the antagonist impedance, which alters the energy output through the port and smooths the force transition. However, the stability of the controllers can be retained only and only if this energy difference is known and, therefore, it can be accounted for in the stability analysis. As detailed in [13], a nonsmooth Lyapunov’s candidate for the system must be laterally unbounded and have a bounded finite derivative. To identify the energy profile, a desired output force profile of the spring during divergence has been selected, following the profile presented in [14]:
(1) 
where is the constant stiffness, is the displacement that triggers the nonlinear spring to activate, indicates the displacement associated with the force saturation, is the saturation force,
is the characteristic length of the sigmoidal function,
controls the saturation velocity ensuring that it happens before with a tolerance greater than of . The energy profile associated with this force profile is as follows:(2) 
where . If these profiles are used in the fractal impedance controller algorithm introduced in [13], they generate a nonsmooth transition while switching from divergence to convergence. The force transition can be smoothed using the Algorithm 1. However, this algorithm introduced a discrepancy in the released energy (Figure 2a), which can be solved via the modulation of the output energy that can be seen as explained earlier as a modulation of the antagonist muscle impedance. The Lyapunov proof is included in Appendix A, which updates the proof given in [13].
IiB Force Feedback
A system interacting with an external entity does not only affect the environment but it is also affected by it. Therefore, the feedforward dynamic interaction enabled by the impedance controller is not sufficient to guarantee the chosen behaviour during interaction. In fact, the environmental impedance will act as a parallel impedance to our controller modifying the resultant behaviour, which implies that we need to adjust to achieve the desired interaction force. An easy to visualise analogue is considering two springs pushing against each other, with one having a tunable stiffness. It is commonly known that the equilibrium point depends on their relative stiffness. Therefore, if we do not alter the exchanged force, we need to adjust the stiffness which also alters the equilibrium point.
An impedance controller interacting with an infinitely rigid system (i.e., undeformable) produces the force at the interface predicted by the selected impedance model. However, the displacement () required to achieve the same force will change when interacting with nonrigid systems, based on their relative stiffness. The parallel Force/Position control was proposed to solve this issue by adjusting the desired position based on a model of the interaction dynamics [17]. The major drawback of that approach was that being applied to a traditional impedance controller, an accurate model and a proper tuning of the impedance controller are essential for retaining the stability. However, the fractal impedance controller does not have such limitations, because the stability condition is embedded into the topology of controller attractor, cf. Fig. 2. This in turn implies that the availability of an accurate expectation of the environment will only enable a more efficient interaction. Thus, an online force feedbackbased haptic exploration (Algorithm 2) can be deployed without introducing any concerns for the controller stability.
Iii Experimental Design
Lastly, it shall be noted that the proposed algorithm is made possible because within the fractal impedance controller the conservative field determined by the stiffness is the only source of energy. Therefore, the only interaction that can alter the topology generated by the controller stiffness is the coupling with an external noninfinite stiffness. On the other hand, the mass and damping component will affect only the path taken for converging to the desired behaviour.
A simulator is a good approach for the preliminary evaluation of the proposed method because it allows direct control of the mechanical properties of the components involved without building a dedicated structure. This allows us to study how changes in properties of the external dynamics affect the controller performance and vice versa. The simulations are realised in Simulink (MathWorks, Inc). The solver is the ode4 with a constant timestep of for continuous contact dynamics. The variable step solver ode23t has been used due to issues encountered in solving the model of the switching contact dynamics. The simulations were run using an Intel i77700HQ CPU with of memory.
The simulation setup is described in Fig. 4, and the system dynamics simulated for . The parameters that have been kept unchanged across all the simulation are , , and
. The implemented hardstop has no transition region for the dynamics and an undamped rebound. The experiments presented in this paper target the study of the influence of the environmental stiffness changes on the controller performance and the evaluation of the effect of contact estimation error on the controller’s performance.
Welded Parallel Connection To SpringDamper Mechanism: The simulation studies a system controlled by the proposed method ”welded” to a springdamper mechanism. The environmental parameters have been set to K and D . The values of K chosen for this simulation are , , , and .
Contact Interaction with a SpringDamper Mechanism: The controller constant stiffness selected for these experiments is K and is equal to the K used in the previous experiment. This simulation is subdivided as follows:

Effect of the change from welded to contact connection changes the system evolution.

Interaction with different values of environmental stiffness. The (K) evaluated are and .

Performance changes introduced by different (D) values. The selected values are and , which is the critical damping for a system with and K .
During the first simulation, the inertial component starts in contact with the hardstop, while it starts at away for the other two simulations. T for the last two experiments have been chosen to be , which is in the linear stiffness region () for the selected parameters.
Iv Results
The results for the Normalised Mean Square Error of the torque tracking error () for the simulations are reported in Fig. 7 and Fig. 8. The MSEs for the welded connection are always contained below 10 of the desired torque even when considering the transient period, and they drop below 10 in the last of the simulated trajectories. However, the introduction of the contact connection shows an increase in to about 10 of the desired torque even at the regime. Indicating that a unilateral connection to an external system is already a source of uncertainties that degrades the performance. The data reported in Fig. 7 indicate that the errors increase even more when introducing the impact with the external environment, wherein some cases they can even reach two times the desired torque.
The time trajectories of the force and position tracking show a transient of about in the case of welded contact (Fig 7), which increases to about when introducing the contact dynamics (Fig 8). The time data analysis for the other experiments shows that despite the high impulsive force generated at the impact, the controller can retain contact in all the simulations, reported in Fig. 9 and Fig. 10. Furthermore, the data show that the torque error is always negative (TT). is still reducing but at an extremely slow rate indicating that may be related to the choice of in Algorithm 2.
V Discussion
The results indicate that the proposed method can safely interact with external dynamics, and it can deal with the impulsive perturbation received when making contact. The controller passivity also enables to modify the controller parameters online without concerns for its stability.
The data also show that some tradeoff needs to be made for obtaining robustness of interaction, particularly when tracking the desired forces. However, other methods are not exempt from similar tradeoffs on the torque tracking accuracy when increasing interaction robustness. Differently from these methods, the proposed controller stability is not associated with the external environment but it is an intrinsic property of the controller. Therefore, it requires fewer assumptions on the environmental dynamics, and if they are violated it degrades the system accuracy but it does not lead to a catastrophic failure of the controller. This inherently safe behaviour is one of the key advantages of our proposed method for the control of realworld robotic systems interacting safely with people and environments.
The integration of Algorithms 1 and 2 into the architecture presented in Fig. 1 has shown that it is possible to reproduce behaviours similar to the one generated by agonist and antagonist muscles in biological systems. This architecture enables the online adaptation of the impedance behaviour without affecting the system stability, which is similar to what has been observed in human by previous studies [8, 10, 18, 19]. The ability to retain the stability during a wide range of unknown dynamic conditions is desirable for applications such as humanrobot interaction (e.g., haptics, rehabilitation robots, and prosthetics), control of cabledriven systems where it may difficult to accurately model the transmission dynamics, and soft robotics. This framework may also find application in modelling human motor control, where the possibility of superimposing multiple controllers may provide a platform to better understand motor synergies.
In conclusion, the proposed method has been proven feasible, and it shows a good level of performance when it is on a bilateral (welded) connection to the environment. The introduction of unilateral contact dynamics increases the force tracking error, however, it does not interfere with the ability of the controller to retain contact and stability of interaction. The residual tracking error seems to be related to the choice of the parameter in Algorithm 2, which should be exposed and optimised by interaction. However, the data also show that this problem can be solved simply by setting . This also enables a higher convergence speed to the desired torque. The passivity of the controller further implies that multiple controllers can be superimposed without interfering with the system stability. Therefore, providing a framework to develop a hierarchical architecture of semiautonomous controllers that can be used to study human motor control and improve robots dynamic interaction performances will be the focus of our future work.
Appendix
Va Lyapunov’s Stability Analysis
The fractal impedance has a nonsmooth piecewise energy manifold with a timeinvariant topology that scales with the controller gains by it does not change shape [13]. Let’s now consider the controller autonomous dynamics for a monodimensional system generated via Algorithm 1:
(3) 
where is the inertia of the system. A valid Lyapunov’s candidate is:
(4) 
where is a constant of energy offset introduced in the switching conditions. Thus, V time derivative is:
(5) 
Therefore, the conditions for stability are respected in both the branches of the system. Nevertheless, being the system nonsmooth to prove stability is needed to verify if V is a Lipschitz function during the switching conditions. Being at the switching conditions due to .
(6) 
The continuity condition can be derived using the following system of equations:
(7) 
being:
and given that the energy contribution of antagonist muscle () always opposes to the motion:
(8) 
which implies:
(9) 
References
 [1] N. Hogan and D. Sternad, “Dynamic primitives of motor behavior,” Biological Cybernetics, vol. 106, no. 1112, pp. 727–739, dec 2012. [Online]. Available: http://link.springer.com/10.1007/s0042201205271
 [2] ——, “Dynamic primitives in the control of locomotion,” Frontiers in computational neuroscience, vol. 7, p. 71, 2013.
 [3] C. Tiseo, K. Veluvolu, and W. Ang, “The bipedal saddle space: modelling and validation,” Bioinspiration & biomimetics, vol. 14, no. 1, p. 015001, 2018.
 [4] C. Tiseo, “Modelling of bipedal locomotion for the development of a compliant pelvic interface between human and a balance assistant robot,” Ph.D. dissertation, Nanyang Technological University, 2018.
 [5] F. Angelini, G. Xin, W. J. Wolfslag, C. Tiseo, M. Mistry, M. Garabini, A. Bicchi, and S. Vijayakumar, “Online optimal impedance planning for legged robots,” in IEEE International Conference on Intelligent Robots and Systems, 2019.
 [6] J. Ahn and N. Hogan, “Walking is not like reaching: Evidence from periodic mechanical perturbations,” PLoS ONE, vol. 7, no. 3, p. e31767, mar 2012. [Online]. Available: http://dx.plos.org/10.1371/journal.pone.0031767
 [7] R. Shadmehr, “Actuator and kinematic redundancy in biological motor control,” in Visual structures and integrated functions. Springer, 1991, pp. 239–254.
 [8] D. W. Franklin, E. Burdet, K. P. Tee, R. Osu, C.M. Chew, T. E. Milner, and M. Kawato, “Cns learns stable, accurate, and efficient movements using a simple algorithm,” Journal of neuroscience, vol. 28, no. 44, pp. 11 165–11 173, 2008.
 [9] G. Ganesh, A. AlbuSchäffer, M. Haruno, M. Kawato, and E. Burdet, “Biomimetic motor behavior for simultaneous adaptation of force, impedance and trajectory in interaction tasks,” in 2010 IEEE International Conference on Robotics and Automation. IEEE, 2010, pp. 2705–2711.
 [10] F. A. MussaIvaldi, N. Hogan, and E. Bizzi, “Neural, mechanical, and geometric factors subserving arm posture in humans.” The Journal of neuroscience : the official journal of the Society for Neuroscience, vol. 5, no. 10, pp. 2732–43, oct 1985.
 [11] R. Shadmehr, “Learning to predict and control the physics of our movements,” Journal of neuroscience, vol. 37, no. 7, pp. 1663–1671, 2017.
 [12] N. Hogan, “A general actuator model based on nonlinear equivalent networks,” IEEE/ASME Transactions on Mechatronics, vol. 19, no. 6, pp. 1929–1939, 2013.
 [13] K. K. Babarahmati, C. Tiseo, J. Smith, H. C. Lin, M. S. Erden, and M. Mistry, “Fractal impedance for passive controllers,” arXiv preprint arXiv:1911.04788, 2019.
 [14] C. Tiseo, W. Merkt, W. Wolfslag, S. Vijayakumar, and M. Mistry, “Safe and compliant control of redundant robots using a stack of passive taskspace controllers,” 2020.
 [15] P. Tommasino and D. Campolo, “An extended passive motion paradigm for humanlike posture and movement planning in redundant manipulators,” Frontiers in neurorobotics, vol. 11, p. 65, 2017.
 [16] S. Schaal, S. Kotosaka, and D. Sternad, “Nonlinear dynamical systems as movement primitives,” in IEEE international conference on humanoid robotics, 2000, pp. 1–11.
 [17] B. Siciliano, “Parallel force/position control of robot manipulators,” in Robotics research. Springer, 1996, pp. 78–89.
 [18] E. Todorov, “Direct cortical control of muscle activation in voluntary arm movements: a model,” Nature neuroscience, vol. 3, no. 4, pp. 391–398, 2000.
 [19] E. Todorov and M. I. Jordan, “Optimal feedback control as a theory of motor coordination,” Nature neuroscience, vol. 5, no. 11, pp. 1226–1235, 2002.