I Introduction
Unmanned Aerial Vehicles (UAV) are being widely used today for applications such as mapping, inspection, exploration and photography [1, 2, 3, 4, 5]. An important observation regarding most applications of aerial robots is that they do not involve any kind of contact with the environment. This phenomenon can be attributed to several challenges. Traditional robotic manipulators whose base links are fixed to the ground (earth) are able to efficiently dissipate external and inertial forces encountered during manipulation. The reaction forces necessary for balancing external as well as inertial forces acting on the manipulator are available immediately owing to the fixed contact relationship of the base link with the infinitely dissipative ground. However, this is not the case for aerial manipulators as multirotor vehicle dynamics are relatively sensitive and “slow” because of inherent aerodynamic forces as well as inertia. Therefore, the task of maintaining the desired vehicle attitude against these external contact forces becomes a challenge. Since multirotors are unstable in general, compensator design and stability for closedloop contact dynamics must be addressed. Along with performing control during contact, the transition between the freeflight regime and the contact regime can present some complications. Physical properties of the UAV platform such as payload, limits on thrust and inertia lead to intricate design challenges.
Researchers have employed a variety of approaches in design and control methodologies in aerial manipulation applications [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Previous research touches on topics such as kinematics for workspace and dexterity, fullbody control, lightweight design, accurate endeffector position control, interaction control, etc. The work in [9] developed a novel parallel manipulator with a large workspace and currentbased torque control to employ impedance control schemes. The work in [6] designed a lightweight 5DoF aerial manipulator for pick and place applications. It has a smart selffolding mechanism to minimize space occupation and static CoG imbalance. A special differential mechanism to cancel attitude disturbances was also designed. The research presented in [11] used deltakinematics for designing a fast and precise aerial manipulator for contactbased inspection. The sophisticated delta structure possesses compliance allowing for slight tracking errors. The researchers in [17] demonstrated the design and operation of a unique superstructure manipulator that has the ability to perch on a vertical surface through impact. It is lightweight and possesses unilateral compliance for perch and release operations.
The authors in [8] developed a 7DoF aerial manipulator for heavy payloads. Hence, it used a backsteppingbased controller for the multirotor, which considers full coupled dynamics and the rapidly shifting CoG. An admittancebased manipulator controller is outlined in the paper. The contribution in [10] presented a multiobjective fullbody controller for the system described in. The fast dynamics of the parallel manipulator ensured efficient kinematic tracking. Furthermore, the work in [16] addressed the problem of interaction in order to track desired contact force using hybrid control for the manipulator, using an impedancebased controller for position and a PI controller for regulating normal force. The authors in [7] presented the design and control of a parallel aerial manipulator for industrial inspection. The approach considered the environment as a compliant contact and used the HuntCrossley interaction model.
This work modeled the NDT (NonDestructive Testing) inspection task as a force regulation problem [7] and designed a lightweight 3DoF RRR manipulator with sufficient dexterity. A planar model of the dynamics during contact was developed, along with a passivitybased PD controller while preserving the nonlinearity of the model. Keeping in mind actuation limits and stability requirements, a simulation of the closedloop dynamics is presented with a smooth freeflight to contact transition scheme. This paper also describes the hardware and software framework for implementation.
Ii Manipulator Design
Although most aerial manipulators have been mounted below the plane of the rotor and t were designed for pick and place tasks. This makes sense spatially as well as for stability. However, since our objective was to achieve continuous contact in the horizontal plane, the manipulator was mounted as a superstructure. Placing the manipulator on top makes the external force a stabilizing moment and also reduces the required manipulator workspace.
Additionally, a static analysis of the system in contact showed that system nonlinearities are more pronounced when the contact force exceeds 6 N. Hence, it was decided to preserve this inherent nonlinearity for the analysis.
Iia Kinematics
The manipulator is an RRR type with the base joint for yaw decoupling of the arm from the multirotor heading. This allows for better error correction during manipulator position tracking. This makes the rest of the arm a planar 2DoF RR type, which provides sufficient dexterity without being too heavy.
The forward kinematics of the manipulator were developed using the DenavitHartenberg (DH) convention. The DH parameters of the manipulator are shown in TABLE I. The inverse kinematics were developed analytically. The first joint angle is obtained using spherical coordinate decoupling. Following this, solving for the inverse kinematics of a 2DoF RR manipulator is trivial.
Transform from link to  

0  
0  0  
0  0 
In the workspace analysis, the joint torque requirements during manipulation were considered since these needed to be minimized to reduce motor size. The relationship was used. Here F is the endeffector force,
is the joint torque vector and
J is the manipulator Jacobian.A typical loadconfiguration combination for this system is with inplane frictional forces equal to about at the configuration , and . The required torques for this configuration from equation (4.14) are computed as , and . The motors were selected by simulating similar such cases.
IiB Compliant Mechanism
Integration of a compliant mechanism at the endeffector was conceived in order to satisfy the following objectives:

To absorb impact when the quadrotor first comes in contact to slow down the resulting contact dynamics.

To allow for slight curvatures on the surface and small errors in the endeffector pose.

To enable compliant forcesensing at the endeffector.
The entire manipulator along with the motors weighed only 327 grams, which was about half of the maximum payload of the quadrotor used for implementation.
Iii Contact Dynamics and Control
The manipulator designed in this work weighed only 327 grams against the platform weight of 2400 g. In addition to this, the maximum shift in CoG for the entire configuration space of the aerial manipulator system was only 2 cm from the vertical axis passing through the centroid of the multirotor. Therefore, no special consideration was given to the effect of the inertial effects of the manipulator with respect to the aerial vehicle in the closedloop dynamics simulations. However, contact introduces an additional external force on the system. The constraints on the thrust values, thrust rates and state variables during flight necessitates a fundamental closedloop dynamics analysis. An important assumption was that the nature of the equation of motion in the vertical direction was considered as static. As contact dynamics are fast compared to that during freeflight, the analysis of the controller performance for this direction is crucial. Hence, this assumption was acceptable. The analysis is performed in a 2D plane and friction has been ignored, compensated by a safety factor.
Applying Newton’s 2 Law, we can write the equations of motion in statespace form. is the moment of inertia about the axis parallel to the yaxis and passing through the CoM of the entire system. For simplicity, the explicit time dependency symbol is dropped hereafter.
(1) 
This equation is in the vectorvalued nonlinear affine (affine in u) form.
Here, x is the state vector and u is the control input.
.
The expression for force from the dynamics can be written as follows:
(2) 
Since the objective was to regulate the contact force, a general approach would have demanded force feedback control from the endeffector. It is indicative from the equation that the regulation of the pitch angle and the xposition implies the regulation of the normal force. Since forcefeedback control (impedance or compliant control) was beyond the scope of this work, this model was used. Thus, the aim was to control the xposition and the pitch angle. The output state to be controlled is expressed by the function . A passivization approach through a nonlinear PD controller was used to model the control law for this system. As the system is affine in u, the control law was derived using the inputoutput linearization technique.
Insisting that the error convergence should follow secondorder dynamics, we can write:
(3) 
In the above equations, and are positive definite matrices containing the gains required for stable dynamics. Using Lie derivatives, the control law was derived.
u  (4) 
The expression for v comes from the second order error dynamics.
(5) 
The energy function can be plotted against time to check for convergence to zero. The energy of the now closedloop system in consideration is given by the sum of the potential and kinetic energies.
(6)  
With appropriate controller gains and initial conditions, the energy equation was shown to converge to zero. From the expression it can be said that the system is stable for any positive damping coefficient.
The initial conditions have a significant influence on the response since the system is nonlinear. These include the xposition , xvelocity , pitch angle and pitch rate . Through appropriate gain selection, the simulation was carried out with conservative limits for the thrust rates. The pitch rate was limited to [15].
Since the contact control mode is forcetriggered, the initial value for force is about Figure (7) (left) shows that steady state is achieved within 0.5 seconds for the force and the pitch angle. The settling time for the xposition is large but that is not relevant as the constantly running inverse kinematics engine of the manipulator will compensate for such errors. Additionally, the thrust values during the transient phase are well below their maximum value of 21 N. Hence, this simulation result for this particular set of initial conditions is satisfactory.
There are certain cases that might occur during transition that may cause the controller to fail i.e. the required control inputs may exceed physical limits. These include cases in which the initial error from the reference values are large enough to cause undesired behaviour in the controller outputs and .

Initial XPosition : This parameter has the least effect on the controller response as the errors associated will not be too large.

Initial XVelocity : Ideally, this is zero as the velocity tends to be small when continuous contact is established. However, certain large positive values tend to reduce the required thrust considerably.

Initial Pitch Angle : If the initial pitch angle reduces to smaller values (below 0.05 rad), the required thrust for satisfactory reference increases rapidly as the contact dynamics are fast compared to the pitching response in figure (7) (right). A value too large would lead to dynamic instability as well as large force values. Since there exists a finite positive value of , a starting value less than the reference is desired. A value of 0.1 rad is chosen since it results in an initial force close to the triggering value and a small overshoot.

Initial Pitch Rate : The worst and the most highly unlikely case is when this is a negative value. This can happen if large impacts create moments in the opposite direction.
According to the above discussion the control would transition smoothly from freeflight regime to contact regime when the following conditions are achieved:

Detection of contact and a threshold force: The endeffector must come into contact with the surface and must possess a threshold force value. To ensure smooth transition and elimination of chatter, a schmitt trigger approach is followed for this condition.

Achieving the desired pose: The pose that agrees with the above discussion as well as the manipulator base joint pose requirements for successful manipulation i.e. that the point on the surface to be applied falls within the workspace and can be manipulated with the desired endeffector orientation.
Iv Implementation
The aerial platform used in this work is a DJI Matrice M100 quadrotor. It has a weight (with battery) of about 2400 grams with a maximum takeoff weight of 3600g. The lightweight nature of the manipulator is the result of clever design optimization as well as high strengthtoweight ratio materials like composites. The servomotor used for link 1 is the MKS DS1220 which is capable of providing 30.4 kg.cm of torque. Link 2, which requires the largest amount of torque, used an MKS HV777 capable of a holding torque of 38 kg.cm. For link 3, a faster and lighter servo was used – Hitec HS77BB.
The onboard processing unit was an Intel NUC i7. It runs a 3.5 GHz processor with 16 GB of RAM. An Arduino Nano microcontroller was used to interface between the servomotors for PWM control as well as to compute the force on the force sensor. It served as a device to receive serial commands from the Intel NUC containing desired manipulator configuration information and to send the value of force back to the NUC. The onboard computer ran ROS (Robotic Operating System) over a Linux OS. Nodes for highlevel quadrotor control using MPC (Model Predictive Control), MCU interfacing, control transition and manipulator inverse kinematics were integrated for the inspection operation.
V Conclusions and Outlook
In conclusion, this work demonstrated the design, controller simulation results and implementation scheme for NDT contact inspection using a three degreeoffreedom aerial manipulator.
One of the most important drawbacks of this work is the assumption of force regulation through pitch and 2D position regulation alone. Thus, a more reliable state observer for force is needed. Such state observers are included in techniques such as torquebased control and compliant control of manipulator interaction. Additionally, the model, although intended to capture the lowlevel nonlinear dynamics, did not consider the effect of the manipulator inertia on the system. From a design perspective, the serial manipulator in this work provides a large workspace and sufficient dexterity. However, a parallel mechanism tends to be faster in kinematic reference tracking along the manipulated surface due to additional actuators. Another point of improvement is the overall design. The offcenter weights may be reduced by mounting all motors along the vertical axis of the quadrotor and use drive mechanisms such as pulleybelts or chainsprockets.
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