In order to model and analyze the position of a trailing camera along a rail system, first principles are used to derive equations of motion of the mass, that can be converted into a transfer functions by method of Laplacians. These transfer functions are then used to simulate the response of a modeled system given a set of inputs. The transfer function that was attained for the camera slider system has voltage as the input and linear position as the output. With the given assumptions and the model created the outcome is reasonable and in accordance with the hypothesis as the final transfer function derived is of a higher order in this case a fourth order type I function. Uncontrolled, the system is unbounded as the final value is observed to be infinity.

Table of Contents

Cameras on rails have been used in cinema since the beginning of film making. It allows for a seamless and smooth capture of a scene’s moving subject matter. These systems have many different configurations, which depend on the intended movement of the scene. Professional camera sliders use electric motors in order to precisely control the movement of the camera during filming. The general system consists of a track, a cart with wheels that mesh with the track, a mounted camera, and a drive system. The drive system can be powered by the wheels or by the track. In this system, four electrically powered motors produce an outputted torque to each wheel through a geared drivetrain.

Figure 1 – Illustration of a camera slider system

Figure 1 illustrates the system that will be analyzed, with as the linear coordinate, being the rotational coordinate, being the torque produced by the DC motors, being the frame mass, which includes the wheels, and being the camera mass.

To achieve these objectives, certain assumptions must be made regarding the system;

· All bodies in the system are rigid.

· Conservative System

· This assumes there are no losses to the surroundings such as heat removed due to frictional losses.

· Lumped mass

· The system masses can be summed to one general mass, this only works if the system is assumed to be symmetrical.

· Fixed ground

· The rails are assumed to be uniform with no deformation.

· Symmetrical system

· Equally balanced shaft/power train, wheels, and body. This allows the system to be divided in four equal parts.

· Gear ratio (1:2)

· Ideal gear to wheel ratio

· Distance between rails is constant

· Ensuring that the rails maintain the same wheel distance ensures that every wheel has the same loading condition.

· Rails are linear not curved

· Keeping the rails linear ensures that every wheel rotates at the same rate during its translation.

· Massless shafts

· This assumes that the mass of shafts is negligible.

· Ideal wheels

· Wheels are perfectly round and have no deformations. This ensures that the ratio between linear displacement and wheel rotation is constant.

· Ideal resistor

· Ideal Inductor

· Fixed field

· Conservative system

· No energy losses to the surroundings in the form of heat or sound.

With the above assumptions, the governing equations of the system can be defined. These equations can be derived using the schematic of the DC motor shown in figure 2 [1].

Figure 2 - Schematic of a DC Motor

To achieve the objectives, the system can be defined in terms of voltage being the input and the produced torque by the DC motor as the output. A transfer function shall aid in the analysis of the higher order differential equations of motion. The ratio of the Laplace of the output and input is the definition of the system’s transfer function and an operator to simulate the output response. An analysis of the circuit and torsional systems is done to achieve this.

The torque produced by the DC motor is directly proportional to the armature current; this can be described as,

Using the lumped mass assumption, the cart can be modeled in such way illustrated in figure 3.

Figure 3 – Simplified mass model

The simplified free body diagram in figure 3 can be analyzed using Newton’s 2nd law; this yield,

The purpose of this project is to analyze a linear actuator with a mounted camera in order to dynamically control the movement of the camera, by using voltage as an input and linear position as the output. The system will be modeled mathematically and will be controlled in Simscape and Simulink.

The objectives for this project are as follows:

· Make reasonable assumptions in order to model the physical system.

· Mathematically model the dynamic system and obtain the necessary transfer functions.

· Create an equivalent system using Simulink.

· Add automatic controls to the system.

· Optimize the control system to obtain desired response output.

Continuing from equation (3), the initial conditions of the second order differential equation are assumed to be equal to zero. Therefore, after some factoring, the Laplacian of equation (3) becomes:

After deriving the transfer function for the system, the equation was coded in MATLAB and following responses were achieved.

Figure 6- Step Response for transfer function

A step response shows the output for a system when the input is changed. In the above defined system, the input is voltage and figure 6 shows how the output behaves once the input (voltage) is changed from 0 to 1.

Figure 7- Impulse response for transfer function

An impulse response shows the change in the output cause some external change and goes to infinity as time changes. Figure 7 shows the impulse response of the linear positioning of our system.

An analysis of the systems response to several different inputs is done to understand the physical response of the mass being modeled. The mass is a cart and camera rolling on a set of rails given the powering torque of a DC motor and gearbox. Here, the input is the DC battery voltage being supplied to the circuit and the output is the translational position of the mass along the rails. Observations are made on the amplitude over time graphs shown in Figures 6 and 7. These graphics illustrate the output over time; that is, the translational position as time passes given either an impulse or a step response.

The step response, in the physical sense, represents an input of voltage that goes from an initial voltage to a final voltage with an infinite rate of change. It jumps from one voltage to another in an instant. Figure 6 shows the position over time given this input. When voltage is increased to a constant DC gain value, the velocity of the motor will increase almost linearly, which in turn increases the rotational position of the motor exponentially. The acceleration can be estimated as constant in this scenario. The position as time approaches infinity appears to be unbounded. The impulse response represented in Figure 7 shows that the position of the cart increases linearly. This implies that velocity is constant. This also signifies that the acceleration is null.

In the case of a steady state system, the input voltage or current does not change with time. The system output is presumed to remain static once a steady state is reached. That is, it reaches a point of equilibrium where there are no variations with time. For the input to be constant implies the motor speed is also constant over time because of the proportional relationship of the voltage and the torque outputted by the motor. If all components are functioning correctly there are no risks other than the physical design of the systems apparatus and the external factors that may impede proper functionality. For example, if electrical equipment such as wires, resistors, inductors, or batteries are not chosen to the correct specifications, the system could fail, create danger, or not function altogether. Risks include, overheating, arching, parasitic current losses, or insufficient power. The dangers associated with these risks may be damage to equipment, property, or even harm to human life in the case of overheating or a rogue mass with an uncontrolled velocity. Apart from the electronic risks, there are mechanical risks in terms of the drivetrain, wheel and rail system. For improper pinion and gear specifications, there exists the risk to expedited wear and tear of the mechanical internals. The associated risk for steady state operation is very low (1 out of 5) since there are no anticipated changes in conditions during steady state operation. To mitigate these risks, proper component selection, proper assembly procedures, and proper fail safes must be put in place in the case of any one component’s cease of function. For the external factors or excitations to the system, a few risk scenarios shall be considered which include impulse, step, and sinusoidal.

For the electro-mechanical system being subject to an impulse, a variety of responses can be observed depending on what is defined as the input and output. It can be an internal or external impulse where the response of the system may have different outcomes. The overarching transfer function can be defined as the ratio of the Laplace of translational position over the voltage input. The torque of the motor is dependent upon the potential difference supplied by the DC battery and effects the angular position of the wheels and then the translational position of the mass. Let the input be the supplied voltage to the circuit and be subject to an impulse. Here, there is a temporary spike in voltage which also corresponds to a proportional spike the torque outputted by the motor. This can cause rapid changes in velocity and acceleration of the wheels, yielding a jerking effect on the rolling mass if no slippage is assumed. This can have risks in terms of increased stresses in mechanical components and shifting of electrical ones. Once the temporary impulse is over, a system of this type typically stabilizes after the impulse. To avoid this sudden acceleration, a limiter can be implemented to limit wheel speed. The risk in this situation can be ranked as a one since the electrical response is much faster than the mechanical response, making the risk associated with the mechanical response negligible.

A sudden change (step function) can cause failure to the system by de-stabilizing the response in terms of linear position. During the time in which the system has a positive voltage input, the linear velocity of the cart will increase linearly, which changes the linear position of the cart exponentially. Instances where the input has a step change can be mitigated with control since it is introduced to the system as error. There are many possible scenarios where some external disturbances can affect the possible output adversely. One of the possibilities is that the camera dolly can derail because of the sudden change in the linear position output and potentially break as well. This kind of phenomenon can happen when the camera dolly impacts with any sort of large physical obstruction in its path. Another possibility of the sudden change in the system can be caused by misalignment in the rails similar to train tracks. This can cause a shift in the camera’s view and can affect the end result of the film. A potential risk can also be the slipping of the camera lobby when it is required to move from one position to another and instead of having a ramp output it has a step output. This can cause the camera dolly to slip off the rail and cause failure. To mitigate these risks, proper inspections should be made to make sure that there are no physical obstructions on track and that the rails are aligned properly. The sudden change in the system’s input or output can cause system instability and that is why it is ranked as medium (3 out of 5).

In the case of fluctuations in the system, this can be both safe and risky, depending on the magnitude and frequency of the sinusoidal input or response. Oscillations in the input and output can be analyzed for risk; more specifically, sinusoidal behavior in the voltage and in the translational position. At low frequencies, this is understood to be part of normal operation since the cart is designed to move a camera along a rail system. In film, it is common to have multiple takes of a scene; therefore, the position will intuitively oscillate. It can be safe for the voltage input to oscillate at relatively low frequencies in the following scenarios: fully reversed oscillation, repeating oscillation, and fluctuating oscillation.

For fully reversed, this entails the repetition of the bidirectional motor rotating forwards and then backwards and the camera moving from point A to point B and back again.

For repeating behavior, the motor is turned on and off repeatedly. In this case the mass can be subject to accelerations to a max velocity and then decelerations to null velocity and then back to a max velocity. For this case in a single direction, the accelerations are not great. For repeating case where there is a change in two directions, the change in direction increases the acceleration and stresses in the system.

Finally, the scenario where there is a fluctuating voltage input results in the very similar behavior to the repeating case. At unusually high frequencies, the risk of equipment or personnel damages increases for all cases. That is why sensing elements such as accelerometers and actuators like rev. limiters are of importance for feedback-based controls. Sinusoidal changes as the input are risky in terms of system stability if the oscillations are sustained, however since lower frequency disturbances are easily controlled, the risk associated with these system changes are relatively low except for higher frequency cases therefor the risk is ranked at 3 out of 5.

References [1] K. Ogata, System Dynamics, Pearson, 2003.

%After deriving the transfer functions, the following is computed.

syms N Jeq b x K m r R L Kb

G1 = (Jeq*x + b)/(m*x*r)

G2 = (1/N)*((((Jeq*x^2+b*x)*(R+L*x))/K)+(Kb*x))^-1 G3 = G1*G2

G4 = simplify(G3)

K = 1 Kb = 1 Jeq = 1 R = 1 r = 1 L = 1 b = 1 N = 1 m = 1TD = simplify(ilaplace(G3,x))num = [1,1] den = [1,2,2,0,0] sys = tf(num,den)step(sys) impulse(sys) bode(sys)

G1 = 

G2 = 

G3 = 

G4 = 

K =

1

Kb =

1

Jeq =

1

R =

1

r =

1

L =

1

b =

1

N =

1

m =

1

Time Domain Function = 

num = 1×2

1 1

den = 1×5

2 2 2 0 0

sys =

s + 1

---------------------

2 s^4 + 2 s^3 + 2 s^2

Continuous-time transfer function.

2