Walking on Mars

Using the Froude equation to develop a predictive model for walking speed for Martian conditions.

Abstract
This project explores the potential walking speed of humans on Mars, a fundamental concern in the broader context of space exploration and Martian colonization. Utilizing experimental data from Earth, we employed the Froude equation to develop a predictive model for walking speeds, which was then adjusted for Martian conditions. Recognizing the inherent uncertainties in modeling, tools such as the Kline-McClintock error propagation method and Monte Carlo simulations were harnessed. Results showed a marked difference in walking speeds on Earth versus Mars, with increased variability on the latter. Confidence intervals were generated to encapsulate this variability, with Mars exhibiting a broader span, indicating more significant uncertainty in predictions. This study exemplifies the confluence of biomechanics and space science, showing light on human adaptability on other planets and paving the way for future biomechanical research in space exploration.

Introduction

Understanding how humans would walk on the Red Planet is fundamental. This question embodies the intersection of planetary science, biomechanics, physiology, and space exploration aspirations.

When considering different gravitational environments, it becomes a complex interplay of muscle function, skeletal support, and neurophysiological feedback. Earth has conditioned our bodies to move a certain way with its unique atmospheric and gravitational properties. In the low-gravity environment of the Moon, astronauts exhibited a hopping gait. With only 38% of Earth’s gravity, Mars presents an entirely different biomechanical puzzle (Cavagna et al., 1998).

However, theoretical models are just the start. Physical experiments, computational simulations, and data from various probes and rovers add layers of complexity to our understanding. Given the resurgence in interest in Mars, fueled by governmental and private endeavors, this locomotion on the Mars question is not merely academic but essential.

Moreover, this quest is about future astronauts and humanity as a species. Understanding our essential functions on another planet becomes foundational as we set our sights on becoming an interplanetary species. Each step on Mars, each astronaut’s stride, will be a testament to human perseverance, curiosity, and our insatiable need to explore. Ultimately, the pace at which we walk on Mars is emblematic of our journey as explorers, innovators, and dreamers.

Background

Mars has been an object of human curiosity for centuries. Our understanding of this celestial body took a giant leap forward during the space age as telescopes improved and space missions were initiated. The culmination of the Moon missions, especially the sight of astronauts adapting their gait to the Moon’s lesser gravity, instigated a new realm of questions about human locomotion on other celestial bodies, particularly Mars.

While rovers like Opportunity, Spirit, and Curiosity provided invaluable data about the Martian landscape, climate, and potential for life, human adaptability remains largely theoretical. Drawing from our lunar experiences, we can make informed hypotheses. The moonwalks of the Apollo missions offered a firsthand view of human movement in a low-gravity environment. The astronauts’ “bunny hops” and bounding strides were adaptations to the Moon’s 1/6th Earth gravity (Ackermann and Van Den Bogert, 2012).

Nevertheless, Mars presents its unique challenges. With a gravity that’s 38% of Earth’s, it strikes a middle ground between the Moon and our home planet. Theoretical models, informed by lunar observations and Earth’s biomechanics, postulate a gait that’s less bounding than the Moon but with longer strides than Earth. However, the reduced Martian gravity is not the only factor. The thin atmosphere, potential for dust storms, and surface terrain variability will all influence human walking (Cavagna et al., 1998).

Recent experiments and simulations, though Earth-bound, have aimed to replicate Martian conditions. Using specially designed treadmills and harness systems that modify effective weight, scientists have begun understanding potential Martian gaits. Preliminary findings suggest increased stride lengths, modified arm swings, and altered postural dynamics. However, these Earth-based replications, while invaluable, have their limitations.

Scope

At the core of our project is the ambition to integrate these fragmented pieces of knowledge into a comprehensive model using deterministic methods. Recognizing that deterministic models offer exact predictions devoid of randomness, we aim to create a model that predicts walking speeds on Mars based on various factors, especially the often-overlooked variability in individual biomechanics like leg length.

Central to our approach is the need for validation. Using the Froude Equation and data gathered from Earth-based experiments, we aim to validate our model extensively. The equation v = T(Fr,g,L) encapsulates the essence of our approach, where v is velocity, Fr is the Froude number, g is the acceleration due to gravity, and L is the leg length. Given the known gravitational acceleration on Mars (g = 3.721m/s2), our primary variable becomes the leg length, which will be sourced from experimental data (Spilker, 2018).

Uncertainty quantification is a critical part of our scope. We recognize that no model, however refined, is immune to errors. We aim to quantify these errors from the experimentally measured leg length and the inherent tool uncertainty due to prediction errors. By propagating uncertainties from both these fronts, we aim to present an expected value for VwalkingonMars and 90% certainty bounds, providing a range of potential walking speeds.

Incorporating advanced techniques like Monte Carlo simulations and the Kline-Mclintock error propagation method, our scope expands to embrace the inherent randomness and variability in experimental data. By simulating thousands of scenarios, the Monte Carlo method will allow us to understand the probable distributions of our outcomes. On the other hand, the Kline-Mclintock method will let us propagate the uncertainties more deterministically, providing a holistic view of potential outcomes.

In conclusion, by combining rigorous scientific methods, experimental data, and simulation techniques, we hope to provide actionable insights into one of the many challenges of Mars colonization.

Figure 1: This image highlights left leg initial contact (IC), foot flat (FF), midstance (MS), heel lift (HL), and toe-off (TO) as significant phases and of importance biomechanically. Foot flat simply refers to forefoot loading, so the entire foot is in ground contact. Image Credit: footbionics.com

Figure 2: Leg Length Measurement with Tape, Image Credit: musculoskeletalkey.com

Method

Understanding how we might traverse its surface becomes essential as we stand on the brink of human exploration of the Red Planet. This study delves deep into data analytics to decode the velocity at which a human might walk on Mars. Our methodology integrates mathematical modeling with empirical data to predict walking speeds while conscientiously factoring in the uncertainties intrinsic to such research.

Our approach will use the foundational Froude equation, which postulates walking speed as a function of gravitational acceleration and leg length. While this equation offers a theoretical point of view, the real-world walking data from Earth breathes life into our models. We gain insights into human gait’s natural variability and patterns by measuring multiple subjects’ leg lengths. Nevertheless, models are rarely perfect mirrors of reality; they come with inherent assumptions and uncertainties. Recognizing this, we employ techniques like Kline-McClintock error propagation and Monte Carlo simulations, ensuring our predictions are mathematically sound and resilient to unpredictable real-world data.

With these mathematical techniques, we lean on advanced data analytics, leveraging histograms, confidence intervals, and speed discrepancy analyses to validate and refine our model. Every step has several validation procedures, from collecting terrestrial walking data to the simulations of Martian walking scenarios. These procedures ascertain the precision of our model by juxtaposing it against empirical data, thereby quantifying the inherent experimental uncertainties.

Model Assumptions and Inherent Uncertainties

In modeling walking velocity on Mars, it is essential to make certain assumptions, and it is crucial to understand the inherent uncertainties associated with these assumptions. Let us consider a simple model that calculates walking velocity as a function of leg length, assuming a constant acceleration due to gravity. Our study revolves around the Froude equation:

v = T(Fr,g,L)

The equation links gait speed, gravitational acceleration, and leg length. The gravitational acceleration is fixed for a particular celestial body. However, variations in individual leg lengths, walking patterns, and other factors create disparities between the model and experimental data. Recognizing and quantifying these deviations is essential. While Mars’ gravity is a fixed value, leg lengths and walking patterns vary across individuals. Furthermore, we have assumed that walking patterns on Mars would mirror those on Earth despite the difference in atmospheric conditions and terrain.

Speed Discrepancies

The biomechanical processes underlying our gait are profoundly ingrained and tailored to Earth’s gravitational force (approximately 99.81m/s2). When we attempt to transpose this Earth-optimized walking mechanism to Mars, whose gravity is about 33.721m/s^2, discrepancies naturally arise.

Firstly, the gravitational force on Mars, being significantly weaker than on Earth, reduces the downward force exerted on the body. This decreased force affects stride length, step frequency, and overall energy exertion. With less gravitational pull, each step can cover a more significant distance, potentially leading to longer strides. However, these longer strides might not translate directly to increased speed, as the cadence or step frequency might be altered due to changes in balance and stability under Mars’s gravity.

Secondly, the human body’s proprioceptive systems, which help us sense our body’s position and motion, have been calibrated for Earth’s conditions. The feedback loops between our muscles, nervous system, and brain, optimized for Earth, might experience a lag or misalignment on Mars. This could result in a different walking pattern or speed as the body adjusts to the new gravitational conditions.

Lastly, the Froude equation, though an excellent baseline, has roots in Earth’s biomechanics. It might only capture the whole picture when applied directly to Mars, considering the physiological and biomechanical adaptations humans undergo in reduced gravity. Factors like muscle atrophy in space, reduced bone density, and potential changes in cardiovascular health could also play a role in influencing the walking speed on Mars.

Confidence Intervals and Validation Variance

When assessing experimental data, particularly in scientific research, it is essential to understand the values we observe and how confident we are about these observations. This confidence hinges on several factors, including measurement error, biological system variability, and our models’ uncertainties.

Confidence Intervals

We have recorded walking speeds on Earth. Due to individual variability in walking speeds because of factors like age, fitness, and biomechanical differences, we observe a range of speeds even for the same leg length. By calculating the CI for this data, we obtain a range that offers insights into the natural variability of walking speeds on Earth.

While we do not have actual observational data for Mars, our simulations would also yield a range of walking speeds, informed by our model and the inherent variability we have learned from Earth data. CIs help define a range within which we expect the actual walking speeds of astronauts on Mars to fall.

Validation Variance

It focuses on the variability in our predictions and highlights the discrepancies between our simulated and observed values. This would indicate how well our biomechanical model captures real-world walking speeds for Earth. A high validation variance would suggest that our model has significant room for improvement, while a low validation variance indicates that the model predictions are close to the observed data.

When extending this concept to Mars, our validation variance would inform us about the potential discrepancies our model might exhibit when applied to Martian conditions. Given that this model is grounded in Earth-based biomechanics and observations, the validation variance for Mars would ideally incorporate adjustments or modifications based on the anticipated physiological and biomechanical changes astronauts would undergo in Mars’ gravity.

This approach ensures that our assessments for Earth and Mars are grounded in statistical rigor, offering a more reliable foundation for any subsequent analyses or predictions.

Experimental Uncertainty

Understanding experimental uncertainty is fundamental to our data-driven exploration. When collecting walking data on Earth, several sources of experimental uncertainty exist. These can stem from variability in the participants’ distractions during data collection, the measurement tools’ precision, or environmental factors like the flooring material and its frictional properties. When we transition from real-world data collection on Earth to simulating walking data on Mars, the landscape of experimental uncertainty shifts significantly. Instead of dealing with the natural variability inherent in human subjects, we grapple with uncertainties in our models, equations, and the parameters we use to simulate Martian conditions. The Froude number, the modeled gravity of Mars, and the biomechanical assumptions we input into our simulations all introduce potential uncertainties. It is worth noting that while real-world data collection has ‘pure error’, simulations possess ‘model uncertainty.’ The latter is the difference between the simulated and actual outcomes in a real Martian environment.

The ‘pure error estimate’ can be obtained for our Earth-based walking data by computing the standard deviation of repeated walking measurements for the same subject. This provides a measure of the spread of the data around the mean, reflecting the inherent variability in the subject’s walking pattern. On the other hand, for Mars, the ‘model uncertainty’ can be estimated by assessing how well our model, adjusted for Martian gravity and other factors, can predict Earth-based walking patterns. Discrepancies here can shed light on potential sources of error that might translate to our Martian simulations.

Kline-McClintock Error Propagation

Error propagation describes how uncertainties in independent variables (input) propagate to the dependent variable (output) in a given function or model. The Kline-McClintock method is a widely accepted approach, especially when addressing multiple sources of uncertainty.

Earth-Based Walking Data

When analyzing walking data on Earth, the primary sources of uncertainty stem from leg length measurements and walking speed. Variability in leg length measurements can result from different measuring techniques, human errors, or even instrument precision. Since our walking model depends on leg length, any uncertainty in this measurement can lead to errors in predicted walking speed. A partial differentiation concerning leg length (L) gives us how changes in (L) affect the walking speed (v).

Mars-Based Simulated Data

The uncertainties associated with the simulated Martian environment are introduced when extrapolating this walking data to Mars. Mars’s gravity is different from Earth’s, which affects walking speed. Additionally, the biomechanical adaptations humans might undergo while walking in reduced gravity scenarios need to be fully understood and modeled.

Using the same Froude equation but with Martian gravity:

v = T(Fr,gmars, L)

Comparing Earth and Mars

It is insightful to compare the sensitivities of walking speed to leg length for both Earth and Mars. This can highlight whether our walking model is more sensitive to leg length variations on Earth versus Mars, guiding us on where to focus our error minimization efforts.

Using the Kline-McClintock error propagation approach, we obtain a clearer picture of how uncertainties in leg length measurements can influence our predictions of walking speed. This is vital for analyzing real-world data on Earth and for the simulations that predict human movement on Mars. By comparing the two, we can gain insights into the robustness of our models and the primary sources of potential discrepancies.

Monte Carlo Simulation for Uncertainty Analysis

Monte Carlo simulations evaluate a range of inputs to provide a probability distribution of potential outcomes. This probabilistic approach gives information and captures inherent variability and uncertainties present in real-world scenarios.

Transition to Mars-Based Simulated Data

When projecting this data onto Mars, gravity is the primary change. The Froude equation models walking speed as a function of leg length and gravity. By applying Monte Carlo simulations, we can consider the uncertainties in our model, leg lengths, and any other parameters we are unsure of, such as biomechanical adaptations humans might undergo in the Martian environment.

Comparative Insights

Comparing the histograms of walking speeds on Earth and Mars, we might observe shifts in the central tendencies or wider spreads in one scenario versus the other. This tells us not only about the effects of gravity but also about the inherent uncertainties when modeling unfamiliar environments like Mars.
The Monte Carlo simulation allows us to harness the power of randomness to study the variability in our data and models. Such simulations offer valuable insights into the probable walking scenarios we might encounter, uncommonly when projecting known experimental data onto an unknown environment like Mars.

Walking Velocity Data Collection on Earth

The transition from walking to running is not arbitrary. It is primarily governed by biomechanics and energy optimization. On Earth, various factors, including the length of one’s legs, determine this transition point. The longer the legs, the larger the step and, often, the higher the speed at which one transitions from walking to running. By gathering data on leg length and walking velocity, we aim to develop a biomechanical model grounded in real-world observations. This data collection becomes our benchmark.

Data Characteristics on Earth

Each entry in our Earth dataset represents a unique combination of leg length and walking velocity. Given that we have twelve entries, it implies we have collected data from 12 distinct individuals. This dataset captures the natural variability among humans. For instance, two individuals with the same leg length might still have different walking speeds due to fitness level, age, or specific biomechanical nuances.

Simulating Walking Data on Mars

When transitioning to Mars, we can only partially take and scale the Earth’s speeds directly. As mentioned earlier, Mars has approximately 38% of Earth’s gravity. This will inevitably affect the biomechanics of walking. By feeding our Earth data into the Froude equation modified for Martian gravity, we can simulate what walking speeds might look like on Mars for the same set of leg lengths.

The Value of Earth Data

With the Earth data as a reference, any simulation for Mars would be plausible. Earth data provides a necessary anchoring point, ensuring our simulations are rooted in reality. It is like understanding walking dynamics in a familiar environment like Earth before predicting it in an unfamiliar one like Mars.

In essence, the walking velocity data collected on Earth is a critical calibration tool for our Mars walking model. It gives us a validation mechanism and the essential parameters to fine-tune our simulations for another planetary environment.

Table 1: Experimental data

Results

Experimental Data Uncertainty

The formulas we utilize are rooted in biomechanics, blending the principles of human movement with the fundamental laws of physics. Central to our calculations is the Froude equation, which ties together the gait speed, gravitational acceleration, and leg length, offering an approximation of walking speed under various gravitational conditions. This equation provides the foundational model for our predictions.

However, every model needs to be revised. We contrast these predictions against real-world data collected from human subjects to understand their accuracy and relevance. The subsequent formulas, ranging from error estimations to experimental uncertainties, serve dual purposes. First, they quantify the divergence between our model’s theoretical outputs and actual human performance. Second, they measure the reliability and variability of our experimental data, ensuring we are not just capturing anomalous behavior but a representative sample of human walking patterns. In essence, these formulas offer a rigorous mathematical lens through which we assess the efficacy and precision of our biomechanical walking model.

Uncertainty in the Model of Walking Velocity

The uncertainty in the walking velocity model is the difference between the velocities predicted by our mathematical model (using the Froude equation) and the velocities we observe in real-world data. This difference indicates how accurate or off-mark our theoretical predictions are compared to real-world walking speeds.

Formula:

Result:

Model Uncertainty: 0.267815 m/s

Estimated Error in the Walking Model

The estimated error quantifies how the predicted velocities from the walking model deviate from actual observed speeds. This is determined by calculating the relative difference between the model’s predictions and the observed data and then averaging over all data points.

Formula:

Result:

Mean Relative Error in Walking Model: 0.134450

Measured Uncertainty in the Experiment

refers to the inherent variability or spread within the collected walking speed data. The standard deviation of observed speeds reflects natural differences in subjects’ walking patterns.

Formula:

Result:

Pure Error Estimate (Measured Uncertainty in the Experiment): 0.240715 m/s

Error in the Walking Model

This is the overall discrepancy between our theoretical predictions from the model and the actual experimental observations. It can be considered an average “miss” of our model over all data points.

Formula:

Result:

Total Error in Walking Model: 0.357953 m/s

Validation Against Experimental Mean

The average walking speed from the model predictions is compared to the average speed from the real-world data. It measures how close our model's "central tendency" is to that of the actual data.

Formula:

Result:

Validation Error Against Experimental Mean: 0.351439 m/s

Experimental Uncertainty

This metric provides an insight into the experimental measurements' spread out or variable. It is given by the coefficient of variation, which is the standard deviation expressed as a percentage of the mean, thus making it a relative measure of dispersion.

Formula:

Result:
Experimental Uncertainty (Coefficient of Variation): 0.094821

Confidence Interval

For the 90% confidence interval (CI) for the mean error in our experimental data, we typically employ the t-distribution since our sample size is relatively small (n < 30). The formula for the confidence interval for the mean of a sample from a normally distributed population when the sample size is small.

The 90% confidence interval for the mean error of our experimental data is plotted, and a histogram of the relative errors is plotted. The histogram lines indicate the mean error and the lower and upper bounds of the 90% confidence interval.

90% CI for Earth walking speeds: [0.091976, 0.17692] m/s

Mars walking simulation

By leveraging models like the Froude equation alongside real-world data on Earth, this simulation will show the difference between our current terrestrial understanding and the impending Martian expeditions.
Kline-McClintock Uncertainty Propagation

The Kline-McClintock method for uncertainty propagation quantifies the uncertainty of an output variable (in this case, walking speed on Mars) based on the uncertainties in the input variables like the Froude number, gravitational acceleration, and leg length.

Formula:

Result:

Monte Carlo Uncertainty Propagation

We use the Monte Carlo simulations to run the model 10,000 times with leg length sampled from their respective probability distributions. After many runs, we will have a distribution of walking speeds, and we can analyze this distribution to understand the variability in the model output due to uncertainties in the inputs.

Considering Uncertainty in Simulation

The relative uncertainty remains constant during the transition from Earth to Mars. This assumption can be made based on the information available for this experiment.

Uncertainty in Leg Length

From the given data, we can calculate the leg length’s standard deviation (or variance) as our measure of uncertainty.

Formula:

Result:
Leg variance = 0.0070

Distribution of Walking Speed on Mars

The Cumulative Distribution Function (CDF) of the potential walking speeds on Mars provides a comprehensive view of the distribution of walking speeds on Mars, helping understand the range, variability, and likelihood of different speeds.

Confidence Interval

We use percentiles from the data to calculate the 90% confidence interval (CI) for the Mars walking speeds obtained from the simulation. The 5th and 95th percentiles represent the bounds of the 90% CI. The plot displays the distribution of the simulated Mars walking speeds along with vertical dashed lines representing the 90% confidence interval. The area between these two dashed lines contains 90% of all simulated speeds.
90% CI for Mars walking speeds: [0.90549, 1.0467] m/s

Discussion

Our analysis revolved around understanding walking speeds on both Earth and Mars. This involved knowing how to interpret, process, and use the experimental data. Recalling the mathematical formulations such as the Froude equation, we tapped into the foundational theories of biomechanics. The CDF allowed us to understand the probability distribution of our walking speeds, offering a comprehensive view of our data. In our models, we quantified various aspects of uncertainty. The model’s mean relative error (0.134450) and the pure error estimate (0.240715 m/s) specifically delineated how our model corresponded to real-world observations. The CDF was pivotal as a foundation for our Monte Carlo simulations guiding our random sampling processes.
The 90% CI for walking speeds on Earth is [0.091976, 0.17692] m/s. This suggests that, based on our Earth data and at a 90% confidence level, we expect the average walking speed to fall within this range. The span of this interval is approximately 0.08494 m/s, a measure of the uncertainty inherent in our Earth-based model.

For Mars, the 90% CI for walking speeds becomes [0.90549, 1.0467] m/s. An immediate observation is that the speeds on Mars, as projected by the model, are significantly higher than those on Earth. This reflects Mars’s different conditions, such as gravity and biomechanical constraints, compared to Earth’s. However, the span of this interval is approximately 0.1412 m/s, which is broader than the Earth’s CI span.

The broader range for Mars suggests increased uncertainty in our Martian predictions compared to our Earth-based ones. It is not surprising given that our Martian model is a projection based on Earth data and involves additional assumptions about conditions on Mars. We need to learn more about walking on Mars than on Earth, which can introduce more variability into the model.

Regarding the model’s performance, we unearthed that the total error in the Earth walking model was 0.357953 m/s. Breaking this down, the validation error against the experimental mean was 0.351439 m/s. The CDF offered a visual representation of this error distribution, allowing the interpretations of where and how our model deviated from experimental results.

The potential walking speeds on Mars are created by combining the insights from our Earth data, Martian constraints, and the probability distributions offered by the CDFs. This allows us a comprehensive view of how terrestrial biomechanics might manifest in an extraterrestrial setting.

The tools and methodologies employed were crucial in offering predictions and establishing these forecasts’ reliability. Future studies could refine this approach by delving into more sophisticated probabilistic models or integrating other biomechanical data sources.

In conclusion, this simulation provides a framework for understanding and predicting human walking speeds on Mars by applying deterministic and probabilistic methods coupled with the insights offered by CDFs. To perfect these models, we need to gather more data and insights. Our predictions will only become more refined and reliable.

Matlab Code

% Given experimental data
data = [0.927 2.778
        0.965 3.148
        0.927 2.380
        0.938 2.288
        1.178 2.364
        1.066 2.450
        1.027 2.717
        0.881 2.610
        0.909 2.380
        0.909 2.427
        0.97 27.475
        1.016 2.448];


leg_length = data(:,1);
observed_speed = data(:,2);
g_earth = 9.80665; % acceleration due to gravity on Earth
Fr = 0.5; % Froude number for walking

% 1. Uncertainty in Model of Walking Velocity
modeled_speed = sqrt(Fr * g_earth * leg_length);
model_uncertainty = std(modeled_speed - observed_speed);

% 2. Estimated Error in the Walking Model
relative_error_model = abs(modeled_speed - observed_speed) ./ observed_speed;
mean_relative_error_model = mean(relative_error_model);

% 3. Measured Uncertainty in the Experiment (Pure Error Estimate)
pure_error_estimate = std(observed_speed);

% 4. Error in the Walking Model
total_model_error = mean(abs(modeled_speed - observed_speed));

% 5. Validation Against Experimental Mean
experimental_mean = mean(observed_speed);
validation_error = abs(experimental_mean - mean(modeled_speed));

% 6. Experimental Uncertainty
experimental_uncertainty = std(observed_speed) / mean(observed_speed); % Coefficient of variation

% Printing the results
fprintf('1. Model Uncertainty: %f m/s\n', model_uncertainty);
fprintf('2. Mean Relative Error in Walking Model: %f\n', mean_relative_error_model);
fprintf('3. Pure Error Estimate (Measured Uncertainty in the Experiment): %f m/s\n', pure_error_estimate);
fprintf('4. Total Error in Walking Model: %f m/s\n', total_model_error);
fprintf('5. Validation Error Against Experimental Mean: %f m/s\n', validation_error);
fprintf('6. Experimental Uncertainty (Coefficient of Variation): %f\n', experimental_uncertainty);



% 1. Uncertainty in Model of Walking Velocity:

% Calculate modeled walking speeds

% Plotting
figure;
plot(leg_length, observed_speed, 'o', leg_length, modeled_speed, '-');
legend('Observed Speed', 'Model Speed');
title('Observed vs. Modeled Walking Speed');
xlabel('Leg Length (m)');
ylabel('Walking Speed (m/s)');


%2. Estimated Error in the Walking Model:

% Histogram of relative error
figure;
histogram(relative_error_model);
title('Histogram of Relative Errors');
xlabel('Relative Error');
ylabel('Frequency');


%3. Measured Uncertainty in the Experiment:

% Histogram of speeds
figure;
histogram(observed_speed);
title('Histogram of Observed Speeds');
xlabel('Speed (m/s)');
ylabel('Frequency');

%4. Error in the Walking Model:

% Plotting error between observed and modeled speeds
figure;
plot(leg_length, observed_speed - modeled_speed, 'o-');
title('Error in Walking Model');
xlabel('Leg Length (m)');
ylabel('Error (m/s)');


%5. Validation Against Experimental Mean:

% Plotting observed vs. mean speed
figure;
plot(leg_length, observed_speed, 'o', leg_length, repmat(experimental_mean, length(leg_length), 1), '--');
legend('Observed Speed', 'Mean Speed');
title('Validation Against Experimental Mean');
xlabel('Leg Length (m)');
ylabel('Walking Speed (m/s)');

%6. Experimental Uncertainty:

% Plotting speed with error bars (standard deviation)
figure;
errorbar(leg_length, observed_speed, pure_error_estimate * ones(size(observed_speed)), 'o');
title('Experimental Uncertainty in Observed Speeds');
xlabel('Leg Length (m)');
ylabel('Walking Speed (m/s)');


% Compute the mean and standard deviation of the errors
mean_error = mean(relative_error_model);
std_error = std(relative_error_model);

% Sample size
n = length(relative_error_model);

% Compute the t critical value for 90% CI
alpha = 0.10;
t_critical = tinv(1 - alpha/2, n-1);

% Compute the margin of error
margin_error = t_critical * (std_error/sqrt(n))

% Confidence Interval
CI_lower = mean_error - margin_error
CI_upper = mean_error + margin_error

disp(['90% CI for Earth walking speeds: [', num2str(CI_lower), ', ', num2str(CI_upper), '] m/s']);

% Plot histogram with mean error and confidence intervals
histogram(error, 'Normalization', 'probability');
hold on;
xline(mean_error, '-r', 'Mean Error');
xline(CI_lower, '--g', 'Lower 90% CI');
xline(CI_upper, '--g', 'Upper 90% CI');
hold off;
title('Histogram of Relative Errors with Mean and 90% CI');
xlabel('Relative Error');
ylabel('Probability');
legend('Error Distribution', 'Mean Error', '90% CI');

% Mars Simulation

g_mars = 3.721;  % Mars gravitational acceleration
Fr = 0.25;

% 1. Kline-McClintock Uncertainty Propagation:
dv_dL = Fr * g_mars / (2 * sqrt(Fr * g_mars * mean(leg_length)))
sigma_L = std(leg_length)
sigma_v = dv_dL * sigma_L % propagated uncertainty

% 2. Monte Carlo Uncertainty Propagation:
num_simulations = 10000;
v_mars_simulated = zeros(num_simulations, 1);

for i = 1:num_simulations
    L_sample = datasample(leg_length, 1);  % sample one leg length from Earth data
    v_mars_simulated(i) = sqrt(Fr * g_mars * L_sample);
end

% Plot histogram of simulated walking speeds on Mars
histogram(v_mars_simulated);
title('Distribution of Walking Speeds on Mars');
xlabel('Walking Speed (m/s)');
ylabel('Frequency');

% Calculate the histogram of v_mars_simulated data with 'Normalization' option set to 'cdf'
[num_counts, bin_edges] = histcounts(v_mars_simulated, 'Normalization', 'cdf');
randomnums = rand(num_simulations, 1);
% The CDF values are represented by 'num_counts', and the corresponding walking speed values (on Mars) are represented by the midpoints of 'bin_edges'
bin_centers = (bin_edges(1:end-1) + bin_edges(2:end)) / 2;


% Plot the CDF
figure;
plot(bin_centers, num_counts, 'LineWidth', 2);
xlabel('Walking Speed on Mars (m/s)');
ylabel('CDF');
title('Cumulative Distribution Function of Walking Speed on Mars');
grid on;


% Calculate the 90% CI
lower_bound = prctile(v_mars_simulated, 5);
upper_bound = prctile(v_mars_simulated, 95);

% Display
disp(['90% CI for Mars walking speeds: [', num2str(lower_bound), ', ', num2str(upper_bound), '] m/s']);

% Plot histogram and the CI
figure;
histogram(v_mars_simulated, 50, 'Normalization', 'pdf'); % Change 50 if you want a different number of bins
hold on;
y_limits = ylim;
plot([lower_bound, lower_bound], y_limits, 'r--', 'LineWidth', 2);
plot([upper_bound, upper_bound], y_limits, 'r--', 'LineWidth', 2);
title('Histogram of Simulated Mars Walking Speeds with 90% CI');
xlabel('Walking Speed (m/s)');
ylabel('Probability Density');
legend('Simulated Speeds', '90% CI');
grid on;
hold off;


% Calculating leg length variance

leg_variance = var(leg_length)
% Simulating walking speed on Mars using the Froude equation
v_mars = sqrt(Fr * g_mars * leg_length);

% Histograms and Plots
figure;
histogram(leg_length);
title('Leg Length Distribution');

References

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