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Modelling and Control of a Virtual Skydiver
Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of 20th Congress Proceedings of the the 20th World World Congress The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International Federation of Control The International Federation of Automatic Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 369–374
Modelling Modelling Modelling
and Control and Control and Control Skydiver Skydiver Skydiver
of of of
a a a
Virtual Virtual Virtual
Anna Clarke ∗∗ Per-Olof Gutman ∗∗ ∗∗ Anna Clarke ∗∗ Per-Olof Gutman ∗∗ Anna Anna Clarke Clarke Per-Olof Per-Olof Gutman Gutman ∗∗ ∗ Technion Autonomous Systems Program, Technion - Israel Institute ∗ ∗ Technion Autonomous Systems Program, Technion - Israel Institute ∗ Technion Autonomous Systems Technion -- Israel of Technology, HaifaProgram, 32000, Israel (e-mail: Technion Autonomous Systems Program, Technion Israel Institute Institute of Technology, Haifa 32000, Israel (e-mail: of Technology, Haifa 32000, Israel (e-mail: [email protected]) of Technology, Haifa 32000, Israel (e-mail: [email protected]) ∗∗ [email protected]) Environmental Engineering, Technion - Israel [email protected]) ∗∗ Faculty of Civil and ∗∗ of Civil and Environmental ∗∗ Faculty Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, Haifa Engineering, 32000, Israel Technion (e-mail: --- Israel Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, Haifa 32000, Israel (e-mail: Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]). Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]). [email protected]). [email protected]). Abstract: An innovative approach of gaining an insight into motor skills involved in human Abstract: innovative approach into skills human Abstract: An innovativeBody approach ofisgaining gaining anofinsight insight into motor motor skills involved instage human body flight An is proposed. flightof the artan maneuvering during theinvolved free fallin of Abstract: An innovative approach of gaining an insight into motor skills involved in human body flight is proposed. Body flight is the art of maneuvering during the free fall stage of body flight is proposed. Body flight is the art of maneuvering during the free fall stage of skydiving, which is a rapidly developing sport. The key idea is creating anthe autonomous system body flight is proposed. Body flight is the art of maneuvering during free fall stage of skydiving, is a rapidly developing sport. key idea is creating an autonomous system skydiving, which is developing sport. The key is an system capable of which performing skydiving maneuvers inThe a virtual way, and turning it into a powerful skydiving, which is a a rapidly rapidly developing sport.in The key idea idea is creating creating an autonomous autonomous system capable of performing skydiving maneuvers a virtual way, and turning it into a powerful capable of performing skydiving maneuvers in a virtual way, and turning it into a powerful tool for improving instruction methods. Towards this goal, the Skydiver Simulator is developed, capable of performing skydiving maneuvers in a virtual way, and turning it into a powerful tool for instruction methods. this the Simulator is tool for improving improving instruction methods. Towards Towards this goal, goal, the Skydiver Skydiver Simulator is developed, developed, comprising Biomechanical, Aerodynamic, and Kinematic Models, Dynamic Equations of Motion, tool for improving instruction methods. Towards this goal, the Skydiver Simulator is developed, comprising Biomechanical, Aerodynamic, and Kinematic Models, Dynamic Equations of Motion, comprising Biomechanical, Aerodynamic, and Kinematic Models, Dynamic Equations of Motion, and a Virtual Reality Environment. A Two-Input-Two-Output controller is designed to track comprising Biomechanical, Aerodynamic, and Kinematic Models, Dynamic Equations of Motion, and a Virtual Reality Environment. A Two-Input-Two-Output controller is designed to track and a Virtual Reality Environment. A Two-Input-Two-Output controller is designed to track the desired inertial motion, producing commands in terms of limbs movements. The natural and a Virtual Reality Environment. A Two-Input-Two-Output controller is designed to track the desired inertial motion, producing commands in terms limbs movements. The natural the desired inertial producing commands in of limbs The natural kinematic redundancy of the human body is resolved by of introducing movement patterns, the desiredredundancy inertial motion, motion, producing commands in terms terms ofintroducing limbs movements. movements. Thepatterns, natural kinematic of the human body is resolved by movement kinematic redundancy redundancy ofergonomic the human human body is is resolved resolved by introducing introducing movement patterns, constructed according toof considerations and empirical knowledge of how skydiving kinematic the body by movement patterns, constructed according to ergonomic considerations and empirical knowledge of how skydiving constructed according to considerations and empirical knowledge of maneuvers are performed. The controller is designed with the use of Quantitative Feedback constructed according to ergonomic ergonomic considerations andwith empirical knowledge of how how skydiving skydiving maneuvers are performed. The controller is designed the use of Quantitative Feedback maneuvers are performed. The controller is designed with the use of Quantitative Feedback Theory, providing robustness for dealing with plant non-linearities and inaccurate execution maneuvers are performed. The controller is designed with the use of Quantitative Feedback Theory, providing robustness dealing with plant non-linearities and inaccurate execution Theory, providing robustness for dealing with non-linearities and execution of the movement patterns. Thefor skydiver simulator allowed reconstruction of many challenging Theory, providingpatterns. robustness for dealingsimulator with plant plant non-linearities and inaccurate inaccurate execution of the movement The skydiver allowed reconstruction of many challenging of the movement patterns. The skydiver simulator allowed reconstruction of many challenging aspects of body-flight observed by practicing skydivers. The virtual skydiver, comprising a of the movement patterns. The skydiver simulator allowed reconstruction of many challenging aspects body-flight observed by practicing skydivers. The virtual skydiver, comprising aspects of body-flight observed by practicing skydivers. The virtual skydiver, comprising controllerof and a guidance algorithm, was shown to autonomously perform meaningful bodyaaa aspects of body-flight observed by practicing skydivers. The virtual skydiver, comprising controller and a guidance algorithm, was shown to autonomously perform meaningful body controller and flight missions. controller and a a guidance guidance algorithm, algorithm, was was shown shown to to autonomously autonomously perform perform meaningful meaningful body body flight missions. flight missions. flight missions. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: autonomous systems, simulators, dynamic modeling, feedback control methods Keywords: Keywords: autonomous autonomous systems, systems, simulators, simulators, dynamic dynamic modeling, modeling, feedback feedback control control methods methods Keywords: autonomous systems, simulators, dynamic modeling, feedback control methods 1. INTRODUCTION ficients were used to analyze stability and control effec1. ficients used analyze stability and control effec1. INTRODUCTION INTRODUCTION ficients were used to topostures. analyze This stability and was control effectiveness were of different analysis compared 1. INTRODUCTION ficients were used to analyze stability and control effectiveness of different postures. This analysis was compared tiveness of different postures. This analysis was compared to basic maneuvers performed in a full-scale vertical wind of different performed postures. This analysis was compared The extensive research on UAVs has covered a great va- tiveness to basic maneuvers in aa using full-scale vertical wind to basic maneuvers performed in full-scale vertical wind tunnel and during actual skydives wearable inertial The extensive research on UAVs has covered a great vabasic maneuvers performed in a using full-scale vertical wind The extensive research on UAVs UAVs has covered covered aerial greatplatva- to riety extensive of remotely controlled and autonomous tunnel during skydives wearable inertial The research on has aa great vatunnel and during actual actualand skydives usingGPS wearable inertial sensors,and magnetometers, miniature loggers. Fullriety of remotely controlled and autonomous aerial plattunnel and during actual skydives using wearable inertial riety of remotely controlled and autonomous aerial platforms, yet one platform remains unresearched to date: the sensors, magnetometers, and miniature GPS loggers. riety ofyet remotely controlled andunresearched autonomous to aerial platsensors, magnetometers, and miniature GPS estimated loggers. FullFullscale aerodynamic forces and moments were by forms, one platform remains date: the sensors, magnetometers, and miniature GPS loggers. Fullforms, yet one platform remains unresearched to date: the human yet body. The problem of skydiver motiontoindate: free-fall aerodynamic forces and moments were estimated by forms, oneThe platform remains unresearched the scale scale aerodynamic forces and moments were estimated by applying a Kalman filter. In Robson and DAndrea (2010) human body. problem of skydiver motion in free-fall aerodynamic forces and moments were estimated by human body. Thethan problem of skydiver skydiver motion in free-fall free-fall is more body. complex stability and control analysis of an scale applying aa Kalman filter. In DAndrea (2010) human The problem of motion in applying Kalman filter.longitudinal In Robson Robson and and DAndrea (2010) the authors conducted stability analysis of is more complex than stability and control analysis of an applying a Kalman filter. In Robson and DAndrea (2010) is more complex than stability and control analysis of an aircraft since a parachutist is a bluff body, is not rigid, authors conducted longitudinal stability analysis of is more complex than stability and control analysis of an the the authors conducted longitudinal stability analysis of a skydiver wearing a jet-powered wingsuit. They used aircraft since a parachutist is a bluff body, is not rigid, the authors conducted longitudinal stability analysis of aircraft since control parachutist is and bluff body, is isdegrees-ofnot rigid, rigid, a skydiver wearing a jet-powered wingsuit. They used aa has multiple surfacesis redundant aircraft since aa parachutist aa bluff body, not arigid skydiver wearing aa jet-powered jet-powered wingsuit. They used aa body assumption and linearized equations of motion has multiple control surfaces and redundant degrees-ofskydiver wearing wingsuit. They used has multiple control surfaces and redundant degrees-offreedom, and performs very complex maneuvers. Whereas arigid body assumption and linearized equations of motion has multiple control surfaces and redundant degrees-ofrigid body assumption and linearized equations of motion similar to those of a glider. Actual flight data was used freedom, and performs very complex maneuvers. Whereas body assumption and linearized equations of motion freedom, and performs very complex maneuvers. Whereas rigid there is aand vastperforms knowledge base for aircraft flight dynamics similar to those of aa glider. Actual flight data was used freedom, very complex maneuvers. Whereas similar to those of glider. Actual flight data was used to verify the theoretical results, using off-the-shelf (Xsens there is a vast knowledge base for aircraft flight dynamics similar to those of a glider. Actual flight data was used there is aa vast vast knowledge base for aircraft aircraft flight dynamics and proved solutions for base automatic control anddynamics various to verify the theoretical results, using off-the-shelf (Xsens there is knowledge for flight to verify the the theoretical theoretical results,suite using off-the-shelf (Xsens Technologies) wearable sensors which held acceleromand solutions for and various verify results, using off-the-shelf (Xsens and proved solutions operation, for automatic automatic control and compavarious to levelsproved of autonomous therecontrol is nothing Technologies) wearable sensors suite which held acceleromand proved solutions for automatic control and various Technologies) wearable sensors suite which held accelerometers, gyros, magnetometers, GPS, and barometers. levels of is wearable sensors suiteand which held acceleromlevels of autonomous autonomous operation, there is nothing nothing comparable for the free-fall operation, parachutist.there In Dietz et al. compa(2011) Technologies) eters, levels of autonomous operation, there is nothing compaeters, gyros, gyros, magnetometers, magnetometers, GPS, GPS, and and barometers. barometers. rable for the free-fall parachutist. In Dietz et al. (2011) eters, gyros, magnetometers, GPS, barometers. rable for the free-fall parachutist. In Dietz et al. (2011) and Myers et al. (2009) an air flow around a free-falling the skydiver rable for the free-fall parachutist. Inaround Dietz et al. (2011) In this paper an analytic method to model and Myers et al. (2009) an air flow a free-falling In this paper an analytic method to model the and Myers et al. al. (2009) (2009) an air air flow flowbyaround around a free-falling free-falling stable parachutist was computed the means of the In In this paper paper an an analytic method to model model the skydiver skydiver aerodynamics is analytic proposed.method It allows a simulation of a and Myers et an a this to the skydiver stable parachutist was computed by means of is proposed. It allows aa simulation of a stable parachutist was to computed by the the into means of the the aerodynamics CFD method in order get an insight problems aerodynamics is proposed. It allows simulation of skydiver continuously altering his body posture and stable parachutist was computed by the means of the aerodynamics is proposed. It allows a simulation of aaa CFD method in order to get an insight into problems skydiver continuously altering his body posture and CFD method in order to get an insight into problems relatedmethod to MC-4 military systeminto deployment. skydiver continuously alteringbasic his skydiving body posture posture and aa controller continuously design implementing maneuvers. CFD in military order to parachute get an insight problems skydiver altering his body and related to system deployment. design implementing basic maneuvers. related to MC-4 MC-4 military parachute system deployment. In Moniuszko et military al. (2010)parachute a skydiver model consisting controller controller design implementing basic skydiving skydiving maneuvers. This means developing an autonomous skydiver - a task related to MC-4 parachute system deployment. controller design implementing basic skydiving maneuvers. In Moniuszko et aa skydiver model means an skydiver -- aaarises task In Moniuszko et al. al. (2010) (2010) skydiver model consisting consisting of 15 simple geometric shapes was developed and posi- This This means developing an autonomous autonomous skydiver task that has not developing been attempted to date. This research In Moniuszko et al. (2010) a skydiver model consisting This means developing an autonomous skydiver - aarises task of 15 simple geometric shapes was developed and posithat has not been attempted to date. This research of 15 simple geometric shapes was developed and positioned in a sit-fly pose.shapes Dynamic movement equations that has not been attempted to date. This research arises not only due to a scientific interest but also due to a vital of 15 simple geometric was developed and posithatonly has due not been attempted to date. This research arises tioned in sit-fly pose. equations scientific but due vital tioned in a sit-fly pose. Dynamic Dynamic movement equations were formulated assuming the body movement was rigid. The aero- not not only due to to aaaproblem scientificofinterest interest but also also due to to a asport: vital and only unresolved the modern skydiving tioned in aa sit-fly pose. Dynamic movement equations not due to scientific interest but also due to a vital were formulated assuming the body was rigid. The aeroand unresolved problem of the modern skydiving sport: were formulated assuming the body was rigid. The aerodynamic coefficients were estimated in a wind tunnel test and unresolved problem of the modern skydiving sport: training of novices. This research aims to incorporate the were formulated assuming the body in was rigid.tunnel The aeroand unresolved problem of the modern skydiving sport: dynamic coefficients were estimated aa wind test training of novices. This research aims to incorporate the dynamic coefficients were estimated in wind tunnel test carried out for three configurations of a skydiver’s manikin. training of novices. This research aims to incorporate the strengths of control engineering and autonomous systems dynamic coefficients were estimated in skydiver’s a wind tunnel test training of novices. This research aims to incorporate the carried for configurations of manikin. of and autonomous carried out foristhree three configurations of a skydiver’s manikin. Similar out work described in Cardona et al. (2011), only strengths strengths of control controltoengineering engineering and learning autonomous systems as a contribution human motor and systems control. carried out for three configurations of aa skydiver’s manikin. strengths of control engineering and autonomous systems Similar work is described in Cardona et al. (2011), only as a contribution to human motor learning and control. Similar work is described in Cardona et al. (2011), only the manikin is positioned the tunnel in belly-to-earth as a contribution to human motor learning and control. Thea first step in this direction is presented here. Similar work is described in Cardona et al. (2011), only as contribution to human motor learning and control. the manikin positioned in in the manikin ismeasured positioneddimensionless in the the tunnel tunnelaerodynamic in belly-to-earth belly-to-earth postures. Theis coef- The The first first step step in in this this direction direction is is presented presented here. here. the manikin is positioned in the tunnel in belly-to-earth The first step in this direction is presented here. postures. The measured dimensionless aerodynamic coefpostures. The measured dimensionless aerodynamic coefpostures. The measured dimensionless aerodynamic coefCopyright © 2017, 2017 IFAC 371 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 371 Copyright © 2017 IFAC 371 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 371Control. 10.1016/j.ifacol.2017.08.160
Proceedings of the 20th IFAC World Congress 370 Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374 Toulouse, France, July 9-14, 2017
2. SKYDIVER MODELING AND SIMULATION 2.1 Biomechanical Model The biomechanical model represents a body in terms of 16 rigid segments and 33 DOF that define the overall body configuration, see Fig. 1. The segments were modeled using simple geometrical shapes and conventional Body Segment Parameters equations (Kwon (1994)) to calculate the local center of gravity and principal moments of inertia for each segment. A rotation quaternion q was computed for each segment defining its orientation relative to the parent segment, thus enabling computation of the overall center of gravity, inertia tensor, and their time derivatives, expressed in the Body coordinate system, arbitrary chosen to be attached to the pelvis joint. For example, (1) shows the transformation chain for right hand, needed for computing its center of gravity in the Body frame RBody cgHand . pperarm Hand = qFHand ⊗ q F orearm ⊗ qTUhorax ⊗ qBody orearm U pperarm
⊗
2 DOF 1 DOF T horax Abdomen qAbdomen ⊗ qBody
(4) Local Limb Frame (5) Local Wind Frame: transformation from Local Limb to Wind frame defines local angles of attack, sideslip, and roll.
3 DOF
3 DOF 3 DOF IN Body Abdomen T horax Abdomen + qAbdomen ⊗ D Hand = qBody ⊗ (D IN T horax U pperarm IN U pperarm IN T horax ⊗ (D U pperarm + qT horax ⊗ (D F orearm + IN F orearm F orearm ))) + qU pperarm ⊗ D Hand
(1)
IN Body Hand RBody + qBody ⊗ Rlocal cgHand = D Hand cgHand
F rame where D IN is the origin of the coordinate system Limb attached to Limb expressed in system F rame.
Conducting similar computations for each segment makes it possible to obtain the body instantaneous center-ofgravity Rcg and inertia tensor I, as shown in (2). NLimbs Body Rcgi mi Rcg = i=1 NLimbs mi i=1 N Limbs Limbi Limbi T I= DCMBody Ilocali (DCMBody ) + (2) i=1 the similarity transf ormation
+
N Limbs i=1
∆Y 2 +∆Z 2 −∆X∆Y −∆X∆Z −∆X∆Y ∆X 2 +∆Z 2 −∆Y ∆Z −∆X∆Z −∆Y ∆Z ∆X 2 +∆Y 2
parallel axis theorem
Fig. 1. Body segments, modeled degrees-of-freedom (DOFs), and coordinate systems
i
mi
where mi - mass of Limb i, [∆X, ∆Y, ∆Z]T - distance between local and global center of gravity, and DCM - direction cosine matrix computed from the relevant quaternion 2.2 Coordinate Systems The following coordinate systems (see Fig. 1) are needed for further computations: (1) Inertial Frame: defined as North, West, Up (2) Body Frame: coincides with Inertial Frame when standing and facing east (3) Global Wind Frame: transformation from Body to Wind frame includes two Euler rotations: α about X-axis, and then −β about Y-axis 372
2.3 Dynamic Equations of Motion The equations of motion were developed according to the Newton-Euler method. The 3D force is the derivative of the linear momentum (3), and the 3D moment is the derivative of the angular momentum (4) Williams (1996). ˙ × rcg + mΩ × rcg ˙ + F = mV˙ + mΩ (3) + Ω × (mV + mΩ × rcg ) ˙ + IΩ ˙ + rcg × mV˙ + rcg ˙ × mV + M = IΩ (4) + V × (mΩ × rcg ) + Ω × (IΩ) + Ω × (rcg × mV ) T
T
where V = [U V W ] - linear velocity, Ω = [P Q R] angular velocity, I - inertia tensor, rcg - center of gravity, all expressed in Body frame, and m - total body mass The inertia tensor and center of gravity vector are constantly changing as the skydiver is altering his body configuration, thus their derivatives may have significant values. 0 −P −Q −R P 0 R −Q I I (5) = 0.5 q q˙Body Q −R 0 P Body R Q −P 0 The inertial orientation of the skydiver is represented by a rotation quaternion, and propagated in time as in (5). 2.4 Kinematic Model The Kinematic Model computes the Euler angles [ψ θ φ] defining the transformation from Inertial to Body frame (6), as well as body angles of attack α and sideslip β (7), and local angles of attack, sideslip, and roll αi , βi , γi of each limb relative to the airflow (8). These angles are required for the Aerodynamic Model. 2(q0 q3 + q1 q2 ) T I qBody = [q0 q1 q2 q3 ] ψ = atan 1 − 2(q22 + q32 ) (6) 2(q0 q1 + q3 q2 ) θ = asin2(q0 q2 − q1 q3 ) φ = atan 1 − 2(q22 + q12 )
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374
α = −atan B qW
V W
β = −asin
U (U 2
2
W 2)
+V + T T α α −β −β = cos sin 0 0 ⊗ cos 0 sin 0 2 2 2 2 Limbi Limbi B qW ind = qBody ⊗ qW = [q0 q1 q2 q3 ] 2(q0 q3 − q1 q2 ) αi = atan 2 q0 − q12 − q22 + q32 βi = −asin2(q0 q2 + q1 q3 ) 2(q0 q3 − q1 q2 ) γi = atan 2 q0 − q22 − q32 + q12
(7)
T
(8)
Limbi B and qW where qW ind are rotation quaternions from Body to Wind and from i-th Limb to Wind frames, respectively.
2.5 Aerodynamic Model The forces and moments acting on the body in free-fall are composed of the gravity and the aerodynamic forces and moments, as summarized in (9) and (10). N limbs 0 I 0 F = Fa i + qBody ⊗ (9) −mg i=1 N limbs i M= rcg × F a i + Ma i + i=1
+ rcg ×
I qBody 2
0 0 ⊗ −mg
−
(10)
− 0.5ρA V HeightCmdamp Ω where g - the gravity constant, ρ - air density, Nlimbs the number of body segments, A - overall area exposed to the airflow, Height - skydiver’s height, Cmdamp i damping coefficients known/estimated, and Fa i , Ma i , rcg - aerodynamic force and moment acting on limb i and its center of gravity expressed in Body frame. The last term in (10) is the aerodynamic damping moment that occurs due to the changes in the orientation of the local wind vector with rotation rates across the skydiver. This moment was measured in wind tunnel experiments in Myers et al. (2009). It is assumed that for a belly-to-earth pose the contribution of the forces and moments of the individual body components can be considered separately. This assumption was inspired by an analogy of flying with an aircraft: the control coefficients due to ailerons and elevators deflection are derived from an analysis of the effect of these actions on a representative aircraft model. In skydiver simulations developed in Nakashima et al. (2004), and Myers et al. (2009) the aerodynamic model is approximated as a sum of forces and moments acting on each individual segment. The experiences of practicing skydivers Works Jr. (1979) also confirms this assumption: skydivers perceive each body component as a separate control surface and focus on the angle at which each limb is presented to the relative wind. The aerodynamic force acting on an individual limb i can be modeled by two components: perpendicular Li and parallel Di to the local wind direction, as shown in (11). 373
371
The aerodynamic moment acting on a body segment which is angled relative to the air flow can be approximated as shown in (14). γi cos 2 2 0.5ρAi V (Clβ )i 0 W ⊗ L i = qB ⊗ 0.5ρAi V 2 (Clα )i 0 (11) 0 γi sin 2 T W D i = qB ⊗ 0 0 0.5ρAreai V 2 Cdmax i where Ai - limb characteristic area (local xz plane), Areai - the total limb area exposed to the airflow approximated according to (12), and (Clα )i , (Clβ )i - the aerodynamic coefficients approximated according to (13), while (Clα )max , (Clβ )max , Cdmax are known/estimated. i i i xz Ai |cosβi sinαi | Areai = max Axy (12) |cosβi cosαi | iyz Ai |sinβi | sin(2αi ) (Clα )i = (Clα )max i (13) max (Clβ )i = (Clβ )i sin(2βi ) γi 2 ρAi V cos l (Cm ) 2 i α i 2 0 W ⊗ Mi = q B ⊗ ρAi V 2 (14) 0 li (Cmβ )i γi 2 sin 0 2 where li - limb characteristic length, and the moment coefficients can be approximated according to (15) , while (Cmα )max , (Cmβ )max are known/estimated i i (Cmα )i = −(Cmα )max sin(2αi ) i (15) sin(2β (Cmβ )i = −(Cmβ )max i) i 2.6 Simulation Tools The overall skydiver model was implemented in Matlab and the graphical environment was written in Virtual Reality Modeling Language. The body configuration of the virtual skydiver can be controlled via a keyboard, resembling a computer game. The simulation was verified by experienced skydivers by moving the limbs of the virtual skydiver and observing the resulting maneuvers. It was found that the simulation reconstructs all basic expected maneuvers in belly-to-earth pose, e.g. deflecting the right arm downwards causes right turn, dropping the thighs causes backslide, straightening the legs causes pitching down and going forward, dropping one knee during turning causes a flip onto the back, and returning the limbs into a neutral position causes stability to be recovered in a belly-to-earth pose. 3. CONTROLLER DESIGN 3.1 Defining Movement Patterns For the purpose of controller design, the individual DOF of the simulated skydiver were organized into movement patterns: combinations of limbs that move synchronously in order to achieve a specific flight movement. Two such patterns were defined: one associated with moving the legs allowing for forward and backward flying, and another one associated with the arms making it possible to turn right and left. Each pattern is defined by
Proceedings of the 20th IFAC World Congress 372 Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374 Toulouse, France, July 9-14, 2017
Fig. 2. Arm and Leg Patterns yaw rate com desired path
F11
+ − Σ
Guidance Algorithm
G11
arm pattern
position orientation
G21 speed com
yaw rate
Fig. 4. Bode Plot P11 : Experienced arm pattern angle to yaw rate
P lant leg
F22
+
Σ −
G22
+ + pattern Σ
speed
Fig. 3. Block Diagram: The plant is given by equations (3),(4) which include equations (9),(10) governed by arm and leg patterns; The controller is given by equations (16),(17). one control signal - the signed amplitude of the deflection angles of the limbs defining the pattern relative to a neutral pose. Fig. 2 shows the arm pattern defined by rotations about [Xlef t , Xright , Zlef t , Zright ]shoulder = [−ν, ν, ν, ν], where ν is the control input u1 (t); and the leg pattern defined by rotations about [Xlef t , Xright ]knee = [µ, µ], [Xlef t , Xright ]hip = [σ, σ], where µ is the control 180 π 2 (−0.0035(µ 180 input u2 (t) and σ = 180 π ) − 0.0335(µ π ) + 17). The patterns were constructed empirically: close to the movements observed in humans.
Fig. 5. Bode Plot P11 : Novice arm pattern angle to yaw rate
3.2 Design Strategy The controller design objective is to track linear and angular velocity commands by the means of two inputs to the non-linear skydiver plant: ’legs’ and ’arms’ patterns defined above. A successful design will allow the skydiver to follow a desired path, translated by a Guidance Algorithm into velocity profiles. Thus, the overall scheme (see Fig. 3) can be used to fulfill a simple and yet meaningful skydiving mission: fly towards another skydiver in the sky and stop in a facing him orientation. The Guidance Algorithm is adopted from guidance for autonomous vehicles, Shmaglit et al. (2006). The controller design starts with constructing transfer functions from each of the movement patterns to yaw rate and velocity, by the means of frequency function analysis (see Figs 4-9). The strategy of designing a TITO controller follows from the strengths of the QFT method Horowitz (1993), Gutman et al. (2007). First, we design the yaw rate loop since the coupling observed in P12 is small (see Fig. 6) and arises only from inaccuracy of the leg pattern execution, i.e. lack of synchronization between left and right leg movements. Representing the cross-coupling by worst-case ’disturbances’, as is done in QFT, will not significantly restrict the design. 374
Fig. 6. Bode Plot P12 : Leg pattern angle to yaw rate. Phase starts from 90 [deg] when the right leg is lagging behind left, and -90 [deg] when the left leg is lagging behind right. In the second design step, whereby the yaw rate loop is already designed, the velocity loop is designed with correct cross-coupling from P21 . The reason for the very strong coupling observed in P21 , see Fig. 7, is that executing the arm pattern induces a backward slide in addition to turning. This effect is well known among skydivers and constitutes a serious challenge for novice jumpers. The design specifications included a zero steady-state error requirement, a servo specification providing the required maneuver agility, and reasonable closed-loop sensitivity and cross-coupling specifications.
Proceedings of the 20th IFAC World Congress Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374 Toulouse, France, July 9-14, 2017
373
Fig. 7. Bode Plot P21 : Experienced arm pattern angle to Body longitudinal velocity
Fig. 10. Nichols Plot: first design step
Fig. 8. Bode Plot P22 : Leg pattern angle to Body longitudinal velocity
pattern, see Fig. 10 and controller (16). This explains why novice skydivers perform turns very slowly and lose stability if they attempt to keep up with more experienced skydivers. s s 0.25(1 + 3.5 ) 1 + 0.7 1 G11 = s s s 1 + 10 1 + 100 (16) s 1 + 0.6 1 F11 = (1 + 7s )(1 + 8s ) 1 + 1s 3.4 Longitudinal Control Loop
Fig. 9. Bode Plot P22 : Leg pattern angle to Inertial horizontal velocity 3.3 Lateral Control Loop Two different ’arm’ patterns were considered for actuating the lateral control loop: one employed by experienced skydivers, defined in Fig. 2, and another one usually observed in novices. The ’novice’ pattern is also defined by four angles: rotation about [Xlef t , Xright , Ylef t , Yright ]shoulder = [−ν, ν, −ν, ν], where ν is the control input.
The analysis of the two patterns from the control theory standpoint reveals an interesting phenomenon, known from skydiving practice but not having until now a theoretical explanation. The ’novice’ transfer function from arm pattern to yaw rate has a resonance - anti-resonance pair around 4 rad/sec (see Fig. 5) that is nonexistent in the ’experienced’ transfer function (see Fig. 4). Due to this fact it was possible to close a considerably higher performance yaw rate tracking loop when utilizing an ’experienced’ 375
The velocity loop design includes two options: tracking the horizontal component of inertial velocity and tracking the longitudinal velocity in the Body frame. The dynamics of the two resulting plants appeared to have significant differences, as shown in Fig.8 and Fig.9. This difference between inertial and body longitudinal velocity is a well known issue among skydivers: initiating forward motion starts with pitching down thus increasing the vertical component of inertial velocity much more than the horizontal component. The ability to reduce the vertical speed component as much as possible, and as fast as possible, is the most crucial skydiving skill, as it is needed for creating separation between skydivers prior to opening the parachutes. Thus, the controller was designed for the harder case: tracking the inertial horizontal speed, see Fig.11. Taking into account that the yaw rate tracking loop designed at step 1 is much faster, it is advantageous to design a precompensator such that G21 reflects our prior knowledge of P21 : a turn induces a backward slide. The design summary is shown in (17). s s 0.1(1 + 1.5 ) 1 + 0.2 1 + 1s G22 = s s s 1 + 0.6 1 + 10 s −0.035(1 + 3s ) 1 + 1s 1 + 0.5 (17) G21 = s s 1 + 5s 1 F22 = (1 + 1s )(1 + 2s ) Notice that the linear controller given by (16),(17) is stabilizing a non-linear plant, which can often be a successful design strategy, Horowitz (1993).
Proceedings of the 20th IFAC World Congress 374 Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374 Toulouse, France, July 9-14, 2017
Fig. 11. Nichols Plot: second design step 4. SIMULATION RESULTS AND FURTHER WORK The controller was applied to the non-linear skydiver simulation in Section 2, and the virtual skydiver successfully completed the mission to fly to another skydiver and stop in front of him, as shown in Fig. 12. The required angles for the two movement patterns, that were computed by the controller during the simulation time, are very small - only a few degrees. This result is consistent with empirical observations of practicing skydivers: only a slight movement of limbs is needed for most of the RW (Relative Work) maneuvers. Adding the controller to the skydiver simulator allowed for tele-operation: ’flying’ the virtual skydiver by the means of yaw rate and speed commands inputted via a keyboard. Analyzing the resulting body postures sequence is the first step towards developing new quantitative and systematic instruction methods. Subsequent steps may include introducing more movement patterns, that prove to be efficient in simulation, thus extending the controller capabilities. Additional maneuvers may include side-sliding, changing vertical speed, barrel rolls, other advanced maneuvers, and even inventing new ones. REFERENCES Cardona, G., Evangelista, D., Ray, N., Tse, K., and Wong, D. (2011). Measurement of the aerodynamic stability and control effectiveness of human skydivers. In American society of biomechanics annual meeting, Long Beach, CA. Dietz, A., Kaszeta, R., Cameron, B., Micka, D., Deserranno, D., and Craley, J. (2011). A cfd toolkit for modeling parachutists in freefall. In 21st AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, 2589. Gutman, P.O., Nordin, M., and Cohen, B. (2007). Recursive grid methods to compute value sets and horowitz– sidi bounds. International Journal of Robust and Nonlinear Control, 17(2-3), 155–171. Horowitz, I.M. (1993). Quantitative feedback design. QFT publications Boulder, CO. Kwon, Y. (1994). Kwon3d motion analysis package 2.1 user’s reference manual. V-TEK Corporation, Anyang, Korea. Moniuszko, J., Maryniak, J., and adyzynska Kozdras, E. (2010). Modelling dynamics and aerodynamic tests of a 376
Fig. 12. Simulation results: the snapshot shows arrival to the final destination sport parachute jumper during flight in sitfly position. Acta of Bioengineering and Biomechanics, 12(3), 29–39. Myers, T.T., Liang, C., and Craley, J. (2009). Free fall analysis and simulation tool (fast). In 20th AIAA Aerodynamic Deccelerator Systems Technology Conference and Seminar, 4–7. Nakashima, M., Aoyama, A., and Omoda, Y. (2004). Development of simulation method for skydiving freefall. The Engineering of Sport 5, 1, 587–593. Robson, G. and DAndrea, R. (2010). Longitudinal stability analysis of a jet-powered wingsuit. In AIAA Atmospheric Flight Mechanics Conference, 7512. Shmaglit, A., Rinat, K., Brand, Z., Fischler, A., and Velger, M. (2006). Autonomous vehicle control and obstacle avoidance concepts oriented to meet the challenging requirements of realistic missions. In 2006 9th International Conference on Control, Automation, Robotics and Vision, 1–6. IEEE. Williams, J.H. (1996). Fundamentals of applied dynamics. New York: John Wiley & Sons, Inc, 1996. Works Jr., M.T.P. (1979). Wings of man - the theory of freefall flight. In 6th AIAA Aerodynamic Decelerator and Balloon Technology Conference, 220. doi: 10.2514/6.1979-452.
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IFAC PapersOnLine 50-1 (2017) 369–374
Modelling Modelling Modelling
and Control and Control and Control Skydiver Skydiver Skydiver
of of of
a a a
Virtual Virtual Virtual
Anna Clarke ∗∗ Per-Olof Gutman ∗∗ ∗∗ Anna Clarke ∗∗ Per-Olof Gutman ∗∗ Anna Anna Clarke Clarke Per-Olof Per-Olof Gutman Gutman ∗∗ ∗ Technion Autonomous Systems Program, Technion - Israel Institute ∗ ∗ Technion Autonomous Systems Program, Technion - Israel Institute ∗ Technion Autonomous Systems Technion -- Israel of Technology, HaifaProgram, 32000, Israel (e-mail: Technion Autonomous Systems Program, Technion Israel Institute Institute of Technology, Haifa 32000, Israel (e-mail: of Technology, Haifa 32000, Israel (e-mail: [email protected]) of Technology, Haifa 32000, Israel (e-mail: [email protected]) ∗∗ [email protected]) Environmental Engineering, Technion - Israel [email protected]) ∗∗ Faculty of Civil and ∗∗ of Civil and Environmental ∗∗ Faculty Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, Haifa Engineering, 32000, Israel Technion (e-mail: --- Israel Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, Haifa 32000, Israel (e-mail: Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]). Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]). [email protected]). [email protected]). Abstract: An innovative approach of gaining an insight into motor skills involved in human Abstract: innovative approach into skills human Abstract: An innovativeBody approach ofisgaining gaining anofinsight insight into motor motor skills involved instage human body flight An is proposed. flightof the artan maneuvering during theinvolved free fallin of Abstract: An innovative approach of gaining an insight into motor skills involved in human body flight is proposed. Body flight is the art of maneuvering during the free fall stage of body flight is proposed. Body flight is the art of maneuvering during the free fall stage of skydiving, which is a rapidly developing sport. The key idea is creating anthe autonomous system body flight is proposed. Body flight is the art of maneuvering during free fall stage of skydiving, is a rapidly developing sport. key idea is creating an autonomous system skydiving, which is developing sport. The key is an system capable of which performing skydiving maneuvers inThe a virtual way, and turning it into a powerful skydiving, which is a a rapidly rapidly developing sport.in The key idea idea is creating creating an autonomous autonomous system capable of performing skydiving maneuvers a virtual way, and turning it into a powerful capable of performing skydiving maneuvers in a virtual way, and turning it into a powerful tool for improving instruction methods. Towards this goal, the Skydiver Simulator is developed, capable of performing skydiving maneuvers in a virtual way, and turning it into a powerful tool for instruction methods. this the Simulator is tool for improving improving instruction methods. Towards Towards this goal, goal, the Skydiver Skydiver Simulator is developed, developed, comprising Biomechanical, Aerodynamic, and Kinematic Models, Dynamic Equations of Motion, tool for improving instruction methods. Towards this goal, the Skydiver Simulator is developed, comprising Biomechanical, Aerodynamic, and Kinematic Models, Dynamic Equations of Motion, comprising Biomechanical, Aerodynamic, and Kinematic Models, Dynamic Equations of Motion, and a Virtual Reality Environment. A Two-Input-Two-Output controller is designed to track comprising Biomechanical, Aerodynamic, and Kinematic Models, Dynamic Equations of Motion, and a Virtual Reality Environment. A Two-Input-Two-Output controller is designed to track and a Virtual Reality Environment. A Two-Input-Two-Output controller is designed to track the desired inertial motion, producing commands in terms of limbs movements. The natural and a Virtual Reality Environment. A Two-Input-Two-Output controller is designed to track the desired inertial motion, producing commands in terms limbs movements. The natural the desired inertial producing commands in of limbs The natural kinematic redundancy of the human body is resolved by of introducing movement patterns, the desiredredundancy inertial motion, motion, producing commands in terms terms ofintroducing limbs movements. movements. Thepatterns, natural kinematic of the human body is resolved by movement kinematic redundancy redundancy ofergonomic the human human body is is resolved resolved by introducing introducing movement patterns, constructed according toof considerations and empirical knowledge of how skydiving kinematic the body by movement patterns, constructed according to ergonomic considerations and empirical knowledge of how skydiving constructed according to considerations and empirical knowledge of maneuvers are performed. The controller is designed with the use of Quantitative Feedback constructed according to ergonomic ergonomic considerations andwith empirical knowledge of how how skydiving skydiving maneuvers are performed. The controller is designed the use of Quantitative Feedback maneuvers are performed. The controller is designed with the use of Quantitative Feedback Theory, providing robustness for dealing with plant non-linearities and inaccurate execution maneuvers are performed. The controller is designed with the use of Quantitative Feedback Theory, providing robustness dealing with plant non-linearities and inaccurate execution Theory, providing robustness for dealing with non-linearities and execution of the movement patterns. Thefor skydiver simulator allowed reconstruction of many challenging Theory, providingpatterns. robustness for dealingsimulator with plant plant non-linearities and inaccurate inaccurate execution of the movement The skydiver allowed reconstruction of many challenging of the movement patterns. The skydiver simulator allowed reconstruction of many challenging aspects of body-flight observed by practicing skydivers. The virtual skydiver, comprising a of the movement patterns. The skydiver simulator allowed reconstruction of many challenging aspects body-flight observed by practicing skydivers. The virtual skydiver, comprising aspects of body-flight observed by practicing skydivers. The virtual skydiver, comprising controllerof and a guidance algorithm, was shown to autonomously perform meaningful bodyaaa aspects of body-flight observed by practicing skydivers. The virtual skydiver, comprising controller and a guidance algorithm, was shown to autonomously perform meaningful body controller and flight missions. controller and a a guidance guidance algorithm, algorithm, was was shown shown to to autonomously autonomously perform perform meaningful meaningful body body flight missions. flight missions. flight missions. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: autonomous systems, simulators, dynamic modeling, feedback control methods Keywords: Keywords: autonomous autonomous systems, systems, simulators, simulators, dynamic dynamic modeling, modeling, feedback feedback control control methods methods Keywords: autonomous systems, simulators, dynamic modeling, feedback control methods 1. INTRODUCTION ficients were used to analyze stability and control effec1. ficients used analyze stability and control effec1. INTRODUCTION INTRODUCTION ficients were used to topostures. analyze This stability and was control effectiveness were of different analysis compared 1. INTRODUCTION ficients were used to analyze stability and control effectiveness of different postures. This analysis was compared tiveness of different postures. This analysis was compared to basic maneuvers performed in a full-scale vertical wind of different performed postures. This analysis was compared The extensive research on UAVs has covered a great va- tiveness to basic maneuvers in aa using full-scale vertical wind to basic maneuvers performed in full-scale vertical wind tunnel and during actual skydives wearable inertial The extensive research on UAVs has covered a great vabasic maneuvers performed in a using full-scale vertical wind The extensive research on UAVs UAVs has covered covered aerial greatplatva- to riety extensive of remotely controlled and autonomous tunnel during skydives wearable inertial The research on has aa great vatunnel and during actual actualand skydives usingGPS wearable inertial sensors,and magnetometers, miniature loggers. Fullriety of remotely controlled and autonomous aerial plattunnel and during actual skydives using wearable inertial riety of remotely controlled and autonomous aerial platforms, yet one platform remains unresearched to date: the sensors, magnetometers, and miniature GPS loggers. riety ofyet remotely controlled andunresearched autonomous to aerial platsensors, magnetometers, and miniature GPS estimated loggers. FullFullscale aerodynamic forces and moments were by forms, one platform remains date: the sensors, magnetometers, and miniature GPS loggers. Fullforms, yet one platform remains unresearched to date: the human yet body. The problem of skydiver motiontoindate: free-fall aerodynamic forces and moments were estimated by forms, oneThe platform remains unresearched the scale scale aerodynamic forces and moments were estimated by applying a Kalman filter. In Robson and DAndrea (2010) human body. problem of skydiver motion in free-fall aerodynamic forces and moments were estimated by human body. Thethan problem of skydiver skydiver motion in free-fall free-fall is more body. complex stability and control analysis of an scale applying aa Kalman filter. In DAndrea (2010) human The problem of motion in applying Kalman filter.longitudinal In Robson Robson and and DAndrea (2010) the authors conducted stability analysis of is more complex than stability and control analysis of an applying a Kalman filter. In Robson and DAndrea (2010) is more complex than stability and control analysis of an aircraft since a parachutist is a bluff body, is not rigid, authors conducted longitudinal stability analysis of is more complex than stability and control analysis of an the the authors conducted longitudinal stability analysis of a skydiver wearing a jet-powered wingsuit. They used aircraft since a parachutist is a bluff body, is not rigid, the authors conducted longitudinal stability analysis of aircraft since control parachutist is and bluff body, is isdegrees-ofnot rigid, rigid, a skydiver wearing a jet-powered wingsuit. They used aa has multiple surfacesis redundant aircraft since aa parachutist aa bluff body, not arigid skydiver wearing aa jet-powered jet-powered wingsuit. They used aa body assumption and linearized equations of motion has multiple control surfaces and redundant degrees-ofskydiver wearing wingsuit. They used has multiple control surfaces and redundant degrees-offreedom, and performs very complex maneuvers. Whereas arigid body assumption and linearized equations of motion has multiple control surfaces and redundant degrees-ofrigid body assumption and linearized equations of motion similar to those of a glider. Actual flight data was used freedom, and performs very complex maneuvers. Whereas body assumption and linearized equations of motion freedom, and performs very complex maneuvers. Whereas rigid there is aand vastperforms knowledge base for aircraft flight dynamics similar to those of aa glider. Actual flight data was used freedom, very complex maneuvers. Whereas similar to those of glider. Actual flight data was used to verify the theoretical results, using off-the-shelf (Xsens there is a vast knowledge base for aircraft flight dynamics similar to those of a glider. Actual flight data was used there is aa vast vast knowledge base for aircraft aircraft flight dynamics and proved solutions for base automatic control anddynamics various to verify the theoretical results, using off-the-shelf (Xsens there is knowledge for flight to verify the the theoretical theoretical results,suite using off-the-shelf (Xsens Technologies) wearable sensors which held acceleromand solutions for and various verify results, using off-the-shelf (Xsens and proved solutions operation, for automatic automatic control and compavarious to levelsproved of autonomous therecontrol is nothing Technologies) wearable sensors suite which held acceleromand proved solutions for automatic control and various Technologies) wearable sensors suite which held accelerometers, gyros, magnetometers, GPS, and barometers. levels of is wearable sensors suiteand which held acceleromlevels of autonomous autonomous operation, there is nothing nothing comparable for the free-fall operation, parachutist.there In Dietz et al. compa(2011) Technologies) eters, levels of autonomous operation, there is nothing compaeters, gyros, gyros, magnetometers, magnetometers, GPS, GPS, and and barometers. barometers. rable for the free-fall parachutist. In Dietz et al. (2011) eters, gyros, magnetometers, GPS, barometers. rable for the free-fall parachutist. In Dietz et al. (2011) and Myers et al. (2009) an air flow around a free-falling the skydiver rable for the free-fall parachutist. Inaround Dietz et al. (2011) In this paper an analytic method to model and Myers et al. (2009) an air flow a free-falling In this paper an analytic method to model the and Myers et al. al. (2009) (2009) an air air flow flowbyaround around a free-falling free-falling stable parachutist was computed the means of the In In this paper paper an an analytic method to model model the skydiver skydiver aerodynamics is analytic proposed.method It allows a simulation of a and Myers et an a this to the skydiver stable parachutist was computed by means of is proposed. It allows aa simulation of a stable parachutist was to computed by the the into means of the the aerodynamics CFD method in order get an insight problems aerodynamics is proposed. It allows simulation of skydiver continuously altering his body posture and stable parachutist was computed by the means of the aerodynamics is proposed. It allows a simulation of aaa CFD method in order to get an insight into problems skydiver continuously altering his body posture and CFD method in order to get an insight into problems relatedmethod to MC-4 military systeminto deployment. skydiver continuously alteringbasic his skydiving body posture posture and aa controller continuously design implementing maneuvers. CFD in military order to parachute get an insight problems skydiver altering his body and related to system deployment. design implementing basic maneuvers. related to MC-4 MC-4 military parachute system deployment. In Moniuszko et military al. (2010)parachute a skydiver model consisting controller controller design implementing basic skydiving skydiving maneuvers. This means developing an autonomous skydiver - a task related to MC-4 parachute system deployment. controller design implementing basic skydiving maneuvers. In Moniuszko et aa skydiver model means an skydiver -- aaarises task In Moniuszko et al. al. (2010) (2010) skydiver model consisting consisting of 15 simple geometric shapes was developed and posi- This This means developing an autonomous autonomous skydiver task that has not developing been attempted to date. This research In Moniuszko et al. (2010) a skydiver model consisting This means developing an autonomous skydiver - aarises task of 15 simple geometric shapes was developed and posithat has not been attempted to date. This research of 15 simple geometric shapes was developed and positioned in a sit-fly pose.shapes Dynamic movement equations that has not been attempted to date. This research arises not only due to a scientific interest but also due to a vital of 15 simple geometric was developed and posithatonly has due not been attempted to date. This research arises tioned in sit-fly pose. equations scientific but due vital tioned in a sit-fly pose. Dynamic Dynamic movement equations were formulated assuming the body movement was rigid. The aero- not not only due to to aaaproblem scientificofinterest interest but also also due to to a asport: vital and only unresolved the modern skydiving tioned in aa sit-fly pose. Dynamic movement equations not due to scientific interest but also due to a vital were formulated assuming the body was rigid. The aeroand unresolved problem of the modern skydiving sport: were formulated assuming the body was rigid. The aerodynamic coefficients were estimated in a wind tunnel test and unresolved problem of the modern skydiving sport: training of novices. This research aims to incorporate the were formulated assuming the body in was rigid.tunnel The aeroand unresolved problem of the modern skydiving sport: dynamic coefficients were estimated aa wind test training of novices. This research aims to incorporate the dynamic coefficients were estimated in wind tunnel test carried out for three configurations of a skydiver’s manikin. training of novices. This research aims to incorporate the strengths of control engineering and autonomous systems dynamic coefficients were estimated in skydiver’s a wind tunnel test training of novices. This research aims to incorporate the carried for configurations of manikin. of and autonomous carried out foristhree three configurations of a skydiver’s manikin. Similar out work described in Cardona et al. (2011), only strengths strengths of control controltoengineering engineering and learning autonomous systems as a contribution human motor and systems control. carried out for three configurations of aa skydiver’s manikin. strengths of control engineering and autonomous systems Similar work is described in Cardona et al. (2011), only as a contribution to human motor learning and control. Similar work is described in Cardona et al. (2011), only the manikin is positioned the tunnel in belly-to-earth as a contribution to human motor learning and control. Thea first step in this direction is presented here. Similar work is described in Cardona et al. (2011), only as contribution to human motor learning and control. the manikin positioned in in the manikin ismeasured positioneddimensionless in the the tunnel tunnelaerodynamic in belly-to-earth belly-to-earth postures. Theis coef- The The first first step step in in this this direction direction is is presented presented here. here. the manikin is positioned in the tunnel in belly-to-earth The first step in this direction is presented here. postures. The measured dimensionless aerodynamic coefpostures. The measured dimensionless aerodynamic coefpostures. The measured dimensionless aerodynamic coefCopyright © 2017, 2017 IFAC 371 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 371 Copyright © 2017 IFAC 371 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 371Control. 10.1016/j.ifacol.2017.08.160
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2. SKYDIVER MODELING AND SIMULATION 2.1 Biomechanical Model The biomechanical model represents a body in terms of 16 rigid segments and 33 DOF that define the overall body configuration, see Fig. 1. The segments were modeled using simple geometrical shapes and conventional Body Segment Parameters equations (Kwon (1994)) to calculate the local center of gravity and principal moments of inertia for each segment. A rotation quaternion q was computed for each segment defining its orientation relative to the parent segment, thus enabling computation of the overall center of gravity, inertia tensor, and their time derivatives, expressed in the Body coordinate system, arbitrary chosen to be attached to the pelvis joint. For example, (1) shows the transformation chain for right hand, needed for computing its center of gravity in the Body frame RBody cgHand . pperarm Hand = qFHand ⊗ q F orearm ⊗ qTUhorax ⊗ qBody orearm U pperarm
⊗
2 DOF 1 DOF T horax Abdomen qAbdomen ⊗ qBody
(4) Local Limb Frame (5) Local Wind Frame: transformation from Local Limb to Wind frame defines local angles of attack, sideslip, and roll.
3 DOF
3 DOF 3 DOF IN Body Abdomen T horax Abdomen + qAbdomen ⊗ D Hand = qBody ⊗ (D IN T horax U pperarm IN U pperarm IN T horax ⊗ (D U pperarm + qT horax ⊗ (D F orearm + IN F orearm F orearm ))) + qU pperarm ⊗ D Hand
(1)
IN Body Hand RBody + qBody ⊗ Rlocal cgHand = D Hand cgHand
F rame where D IN is the origin of the coordinate system Limb attached to Limb expressed in system F rame.
Conducting similar computations for each segment makes it possible to obtain the body instantaneous center-ofgravity Rcg and inertia tensor I, as shown in (2). NLimbs Body Rcgi mi Rcg = i=1 NLimbs mi i=1 N Limbs Limbi Limbi T I= DCMBody Ilocali (DCMBody ) + (2) i=1 the similarity transf ormation
+
N Limbs i=1
∆Y 2 +∆Z 2 −∆X∆Y −∆X∆Z −∆X∆Y ∆X 2 +∆Z 2 −∆Y ∆Z −∆X∆Z −∆Y ∆Z ∆X 2 +∆Y 2
parallel axis theorem
Fig. 1. Body segments, modeled degrees-of-freedom (DOFs), and coordinate systems
i
mi
where mi - mass of Limb i, [∆X, ∆Y, ∆Z]T - distance between local and global center of gravity, and DCM - direction cosine matrix computed from the relevant quaternion 2.2 Coordinate Systems The following coordinate systems (see Fig. 1) are needed for further computations: (1) Inertial Frame: defined as North, West, Up (2) Body Frame: coincides with Inertial Frame when standing and facing east (3) Global Wind Frame: transformation from Body to Wind frame includes two Euler rotations: α about X-axis, and then −β about Y-axis 372
2.3 Dynamic Equations of Motion The equations of motion were developed according to the Newton-Euler method. The 3D force is the derivative of the linear momentum (3), and the 3D moment is the derivative of the angular momentum (4) Williams (1996). ˙ × rcg + mΩ × rcg ˙ + F = mV˙ + mΩ (3) + Ω × (mV + mΩ × rcg ) ˙ + IΩ ˙ + rcg × mV˙ + rcg ˙ × mV + M = IΩ (4) + V × (mΩ × rcg ) + Ω × (IΩ) + Ω × (rcg × mV ) T
T
where V = [U V W ] - linear velocity, Ω = [P Q R] angular velocity, I - inertia tensor, rcg - center of gravity, all expressed in Body frame, and m - total body mass The inertia tensor and center of gravity vector are constantly changing as the skydiver is altering his body configuration, thus their derivatives may have significant values. 0 −P −Q −R P 0 R −Q I I (5) = 0.5 q q˙Body Q −R 0 P Body R Q −P 0 The inertial orientation of the skydiver is represented by a rotation quaternion, and propagated in time as in (5). 2.4 Kinematic Model The Kinematic Model computes the Euler angles [ψ θ φ] defining the transformation from Inertial to Body frame (6), as well as body angles of attack α and sideslip β (7), and local angles of attack, sideslip, and roll αi , βi , γi of each limb relative to the airflow (8). These angles are required for the Aerodynamic Model. 2(q0 q3 + q1 q2 ) T I qBody = [q0 q1 q2 q3 ] ψ = atan 1 − 2(q22 + q32 ) (6) 2(q0 q1 + q3 q2 ) θ = asin2(q0 q2 − q1 q3 ) φ = atan 1 − 2(q22 + q12 )
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374
α = −atan B qW
V W
β = −asin
U (U 2
2
W 2)
+V + T T α α −β −β = cos sin 0 0 ⊗ cos 0 sin 0 2 2 2 2 Limbi Limbi B qW ind = qBody ⊗ qW = [q0 q1 q2 q3 ] 2(q0 q3 − q1 q2 ) αi = atan 2 q0 − q12 − q22 + q32 βi = −asin2(q0 q2 + q1 q3 ) 2(q0 q3 − q1 q2 ) γi = atan 2 q0 − q22 − q32 + q12
(7)
T
(8)
Limbi B and qW where qW ind are rotation quaternions from Body to Wind and from i-th Limb to Wind frames, respectively.
2.5 Aerodynamic Model The forces and moments acting on the body in free-fall are composed of the gravity and the aerodynamic forces and moments, as summarized in (9) and (10). N limbs 0 I 0 F = Fa i + qBody ⊗ (9) −mg i=1 N limbs i M= rcg × F a i + Ma i + i=1
+ rcg ×
I qBody 2
0 0 ⊗ −mg
−
(10)
− 0.5ρA V HeightCmdamp Ω where g - the gravity constant, ρ - air density, Nlimbs the number of body segments, A - overall area exposed to the airflow, Height - skydiver’s height, Cmdamp i damping coefficients known/estimated, and Fa i , Ma i , rcg - aerodynamic force and moment acting on limb i and its center of gravity expressed in Body frame. The last term in (10) is the aerodynamic damping moment that occurs due to the changes in the orientation of the local wind vector with rotation rates across the skydiver. This moment was measured in wind tunnel experiments in Myers et al. (2009). It is assumed that for a belly-to-earth pose the contribution of the forces and moments of the individual body components can be considered separately. This assumption was inspired by an analogy of flying with an aircraft: the control coefficients due to ailerons and elevators deflection are derived from an analysis of the effect of these actions on a representative aircraft model. In skydiver simulations developed in Nakashima et al. (2004), and Myers et al. (2009) the aerodynamic model is approximated as a sum of forces and moments acting on each individual segment. The experiences of practicing skydivers Works Jr. (1979) also confirms this assumption: skydivers perceive each body component as a separate control surface and focus on the angle at which each limb is presented to the relative wind. The aerodynamic force acting on an individual limb i can be modeled by two components: perpendicular Li and parallel Di to the local wind direction, as shown in (11). 373
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The aerodynamic moment acting on a body segment which is angled relative to the air flow can be approximated as shown in (14). γi cos 2 2 0.5ρAi V (Clβ )i 0 W ⊗ L i = qB ⊗ 0.5ρAi V 2 (Clα )i 0 (11) 0 γi sin 2 T W D i = qB ⊗ 0 0 0.5ρAreai V 2 Cdmax i where Ai - limb characteristic area (local xz plane), Areai - the total limb area exposed to the airflow approximated according to (12), and (Clα )i , (Clβ )i - the aerodynamic coefficients approximated according to (13), while (Clα )max , (Clβ )max , Cdmax are known/estimated. i i i xz Ai |cosβi sinαi | Areai = max Axy (12) |cosβi cosαi | iyz Ai |sinβi | sin(2αi ) (Clα )i = (Clα )max i (13) max (Clβ )i = (Clβ )i sin(2βi ) γi 2 ρAi V cos l (Cm ) 2 i α i 2 0 W ⊗ Mi = q B ⊗ ρAi V 2 (14) 0 li (Cmβ )i γi 2 sin 0 2 where li - limb characteristic length, and the moment coefficients can be approximated according to (15) , while (Cmα )max , (Cmβ )max are known/estimated i i (Cmα )i = −(Cmα )max sin(2αi ) i (15) sin(2β (Cmβ )i = −(Cmβ )max i) i 2.6 Simulation Tools The overall skydiver model was implemented in Matlab and the graphical environment was written in Virtual Reality Modeling Language. The body configuration of the virtual skydiver can be controlled via a keyboard, resembling a computer game. The simulation was verified by experienced skydivers by moving the limbs of the virtual skydiver and observing the resulting maneuvers. It was found that the simulation reconstructs all basic expected maneuvers in belly-to-earth pose, e.g. deflecting the right arm downwards causes right turn, dropping the thighs causes backslide, straightening the legs causes pitching down and going forward, dropping one knee during turning causes a flip onto the back, and returning the limbs into a neutral position causes stability to be recovered in a belly-to-earth pose. 3. CONTROLLER DESIGN 3.1 Defining Movement Patterns For the purpose of controller design, the individual DOF of the simulated skydiver were organized into movement patterns: combinations of limbs that move synchronously in order to achieve a specific flight movement. Two such patterns were defined: one associated with moving the legs allowing for forward and backward flying, and another one associated with the arms making it possible to turn right and left. Each pattern is defined by
Proceedings of the 20th IFAC World Congress 372 Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374 Toulouse, France, July 9-14, 2017
Fig. 2. Arm and Leg Patterns yaw rate com desired path
F11
+ − Σ
Guidance Algorithm
G11
arm pattern
position orientation
G21 speed com
yaw rate
Fig. 4. Bode Plot P11 : Experienced arm pattern angle to yaw rate
P lant leg
F22
+
Σ −
G22
+ + pattern Σ
speed
Fig. 3. Block Diagram: The plant is given by equations (3),(4) which include equations (9),(10) governed by arm and leg patterns; The controller is given by equations (16),(17). one control signal - the signed amplitude of the deflection angles of the limbs defining the pattern relative to a neutral pose. Fig. 2 shows the arm pattern defined by rotations about [Xlef t , Xright , Zlef t , Zright ]shoulder = [−ν, ν, ν, ν], where ν is the control input u1 (t); and the leg pattern defined by rotations about [Xlef t , Xright ]knee = [µ, µ], [Xlef t , Xright ]hip = [σ, σ], where µ is the control 180 π 2 (−0.0035(µ 180 input u2 (t) and σ = 180 π ) − 0.0335(µ π ) + 17). The patterns were constructed empirically: close to the movements observed in humans.
Fig. 5. Bode Plot P11 : Novice arm pattern angle to yaw rate
3.2 Design Strategy The controller design objective is to track linear and angular velocity commands by the means of two inputs to the non-linear skydiver plant: ’legs’ and ’arms’ patterns defined above. A successful design will allow the skydiver to follow a desired path, translated by a Guidance Algorithm into velocity profiles. Thus, the overall scheme (see Fig. 3) can be used to fulfill a simple and yet meaningful skydiving mission: fly towards another skydiver in the sky and stop in a facing him orientation. The Guidance Algorithm is adopted from guidance for autonomous vehicles, Shmaglit et al. (2006). The controller design starts with constructing transfer functions from each of the movement patterns to yaw rate and velocity, by the means of frequency function analysis (see Figs 4-9). The strategy of designing a TITO controller follows from the strengths of the QFT method Horowitz (1993), Gutman et al. (2007). First, we design the yaw rate loop since the coupling observed in P12 is small (see Fig. 6) and arises only from inaccuracy of the leg pattern execution, i.e. lack of synchronization between left and right leg movements. Representing the cross-coupling by worst-case ’disturbances’, as is done in QFT, will not significantly restrict the design. 374
Fig. 6. Bode Plot P12 : Leg pattern angle to yaw rate. Phase starts from 90 [deg] when the right leg is lagging behind left, and -90 [deg] when the left leg is lagging behind right. In the second design step, whereby the yaw rate loop is already designed, the velocity loop is designed with correct cross-coupling from P21 . The reason for the very strong coupling observed in P21 , see Fig. 7, is that executing the arm pattern induces a backward slide in addition to turning. This effect is well known among skydivers and constitutes a serious challenge for novice jumpers. The design specifications included a zero steady-state error requirement, a servo specification providing the required maneuver agility, and reasonable closed-loop sensitivity and cross-coupling specifications.
Proceedings of the 20th IFAC World Congress Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374 Toulouse, France, July 9-14, 2017
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Fig. 7. Bode Plot P21 : Experienced arm pattern angle to Body longitudinal velocity
Fig. 10. Nichols Plot: first design step
Fig. 8. Bode Plot P22 : Leg pattern angle to Body longitudinal velocity
pattern, see Fig. 10 and controller (16). This explains why novice skydivers perform turns very slowly and lose stability if they attempt to keep up with more experienced skydivers. s s 0.25(1 + 3.5 ) 1 + 0.7 1 G11 = s s s 1 + 10 1 + 100 (16) s 1 + 0.6 1 F11 = (1 + 7s )(1 + 8s ) 1 + 1s 3.4 Longitudinal Control Loop
Fig. 9. Bode Plot P22 : Leg pattern angle to Inertial horizontal velocity 3.3 Lateral Control Loop Two different ’arm’ patterns were considered for actuating the lateral control loop: one employed by experienced skydivers, defined in Fig. 2, and another one usually observed in novices. The ’novice’ pattern is also defined by four angles: rotation about [Xlef t , Xright , Ylef t , Yright ]shoulder = [−ν, ν, −ν, ν], where ν is the control input.
The analysis of the two patterns from the control theory standpoint reveals an interesting phenomenon, known from skydiving practice but not having until now a theoretical explanation. The ’novice’ transfer function from arm pattern to yaw rate has a resonance - anti-resonance pair around 4 rad/sec (see Fig. 5) that is nonexistent in the ’experienced’ transfer function (see Fig. 4). Due to this fact it was possible to close a considerably higher performance yaw rate tracking loop when utilizing an ’experienced’ 375
The velocity loop design includes two options: tracking the horizontal component of inertial velocity and tracking the longitudinal velocity in the Body frame. The dynamics of the two resulting plants appeared to have significant differences, as shown in Fig.8 and Fig.9. This difference between inertial and body longitudinal velocity is a well known issue among skydivers: initiating forward motion starts with pitching down thus increasing the vertical component of inertial velocity much more than the horizontal component. The ability to reduce the vertical speed component as much as possible, and as fast as possible, is the most crucial skydiving skill, as it is needed for creating separation between skydivers prior to opening the parachutes. Thus, the controller was designed for the harder case: tracking the inertial horizontal speed, see Fig.11. Taking into account that the yaw rate tracking loop designed at step 1 is much faster, it is advantageous to design a precompensator such that G21 reflects our prior knowledge of P21 : a turn induces a backward slide. The design summary is shown in (17). s s 0.1(1 + 1.5 ) 1 + 0.2 1 + 1s G22 = s s s 1 + 0.6 1 + 10 s −0.035(1 + 3s ) 1 + 1s 1 + 0.5 (17) G21 = s s 1 + 5s 1 F22 = (1 + 1s )(1 + 2s ) Notice that the linear controller given by (16),(17) is stabilizing a non-linear plant, which can often be a successful design strategy, Horowitz (1993).
Proceedings of the 20th IFAC World Congress 374 Anna Clarke et al. / IFAC PapersOnLine 50-1 (2017) 369–374 Toulouse, France, July 9-14, 2017
Fig. 11. Nichols Plot: second design step 4. SIMULATION RESULTS AND FURTHER WORK The controller was applied to the non-linear skydiver simulation in Section 2, and the virtual skydiver successfully completed the mission to fly to another skydiver and stop in front of him, as shown in Fig. 12. The required angles for the two movement patterns, that were computed by the controller during the simulation time, are very small - only a few degrees. This result is consistent with empirical observations of practicing skydivers: only a slight movement of limbs is needed for most of the RW (Relative Work) maneuvers. Adding the controller to the skydiver simulator allowed for tele-operation: ’flying’ the virtual skydiver by the means of yaw rate and speed commands inputted via a keyboard. Analyzing the resulting body postures sequence is the first step towards developing new quantitative and systematic instruction methods. Subsequent steps may include introducing more movement patterns, that prove to be efficient in simulation, thus extending the controller capabilities. Additional maneuvers may include side-sliding, changing vertical speed, barrel rolls, other advanced maneuvers, and even inventing new ones. REFERENCES Cardona, G., Evangelista, D., Ray, N., Tse, K., and Wong, D. (2011). Measurement of the aerodynamic stability and control effectiveness of human skydivers. In American society of biomechanics annual meeting, Long Beach, CA. Dietz, A., Kaszeta, R., Cameron, B., Micka, D., Deserranno, D., and Craley, J. (2011). A cfd toolkit for modeling parachutists in freefall. In 21st AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, 2589. Gutman, P.O., Nordin, M., and Cohen, B. (2007). Recursive grid methods to compute value sets and horowitz– sidi bounds. International Journal of Robust and Nonlinear Control, 17(2-3), 155–171. Horowitz, I.M. (1993). Quantitative feedback design. QFT publications Boulder, CO. Kwon, Y. (1994). Kwon3d motion analysis package 2.1 user’s reference manual. V-TEK Corporation, Anyang, Korea. Moniuszko, J., Maryniak, J., and adyzynska Kozdras, E. (2010). Modelling dynamics and aerodynamic tests of a 376
Fig. 12. Simulation results: the snapshot shows arrival to the final destination sport parachute jumper during flight in sitfly position. Acta of Bioengineering and Biomechanics, 12(3), 29–39. Myers, T.T., Liang, C., and Craley, J. (2009). Free fall analysis and simulation tool (fast). In 20th AIAA Aerodynamic Deccelerator Systems Technology Conference and Seminar, 4–7. Nakashima, M., Aoyama, A., and Omoda, Y. (2004). Development of simulation method for skydiving freefall. The Engineering of Sport 5, 1, 587–593. Robson, G. and DAndrea, R. (2010). Longitudinal stability analysis of a jet-powered wingsuit. In AIAA Atmospheric Flight Mechanics Conference, 7512. Shmaglit, A., Rinat, K., Brand, Z., Fischler, A., and Velger, M. (2006). Autonomous vehicle control and obstacle avoidance concepts oriented to meet the challenging requirements of realistic missions. In 2006 9th International Conference on Control, Automation, Robotics and Vision, 1–6. IEEE. Williams, J.H. (1996). Fundamentals of applied dynamics. New York: John Wiley & Sons, Inc, 1996. Works Jr., M.T.P. (1979). Wings of man - the theory of freefall flight. In 6th AIAA Aerodynamic Decelerator and Balloon Technology Conference, 220. doi: 10.2514/6.1979-452.